CONTENTS IN BRIEF
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
Business Snapshots
Preface
Introduction
Financial Products and How They are Used for Hedging
How Traders Manage Their Exposures
Interest Rate Risk
Volatility
Correlation and Copulas
Bank Regulation and Basel II
The VaR Measure
Market Risk VaR: Historical Simulation Approach
Market Risk VaR: Model-Building Approach
Credit Risk: Estimating Default Probabilities
Credit Risk Losses and Credit VaR
Credit Derivatives
Operational Risk
Model Risk and Liquidity Risk
Economic Capital and RAROC
Weather, Energy, and Insurance Derivatives
Big Losses and What We Can Learn from Them
Appendix A: Valuing Forward and Futures Contracts
Appendix B: Valuing Swaps
Appendix C: Valuing European Options
Appendix D: Valuing American Options
Appendix E: Manipulation of Credit Transition Matrices
Answers to Questions and Problems
Glossary of Terms
Derivagem Software
Tables for N(x)
Index
xiii
xv
1
27
55
79
1ll
143
165
195
217
233
255
277
299
321
343
365
385
395
407
409
413
417
421
423
457
479
484
487
Contents
Business Snapshots
Preface
Chapter 1
xiii
xv
Introduction
1.1
Risk vs. return for investors
1.2
Risk vs. return for companies
1.3
Bank capital
1.4
Approaches to managing risk
1.5
The management of net interest income
Summary
Further reading
Questions and problems
Assignment questions
Chapter 2 Financial products and how they are used for hedging
2.1
The markets
2.2
When to hedge
2.3
The "plain vanilla" products
2.4
Using the products for hedging
2.5
Exotic options and structured deals
2.6
Dangers
Summary
Further reading
Questions and problems
Assignment questions
1
2
12
15
18
20
22
23
24
25
27
27
28
30
43
46
48
48
50
50
53
Chapter 3 How
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
55
55
63
65
67
69
69
70
72
traders manage their exposures
Delta
Gamma
Vega
Theta
Rho
Calculating Greek letters
Taylor series expansions
The realities of hedging
Contents
viii
3.9
3.10
Hedging exotics
Scenario analysis
Summary
Further reading
Questions and problems
Assignment questions
73
74
75
76
76
78
Chapter 4 Interest rate risk
4.1
Measuring interest rates
4.2
Zero rates and forward rates
4.3
Treasury rates
4.4
LIBOR and swap rates
4.5
Duration
4.6
Convexity
4.7
Application to portfolios
4.8
Nonparallel yield curve shifts
4.9
Interest rate deltas
4.10 Principal components analysis
4.11 Gamma and vega
Summary
Further reading
Questions and problems
Assignment questions
79
80
83
85
87
89
93
94
96
98
100
104
105
106
106
108
Chapter
111
112
114
115
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
Chapter 6
5
Volatility
Definition of volatility
Implied volatilities
Estimating volatility from historical data
Are daily percentage changes in financial variables
normal?
Monitoring daily volatility
The exponentially weighted moving average model
The GARCH(1,1) model
Choosing between the models
Maximum-likelihood methods
Using GARCH(1,1) to forecast future volatility
Summary
Further reading
Questions and problems
Assignment questions
Correlations and Copulas
6.1
Definition of correlation
6.2
Monitoring correlation
6.3
Multivariate normal distributions
117
121
123
125
127
127
133
137
138
139
140
143
144
146
149
Contents
ix
6.4
6.5
Chapter 7 Bank
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
' 7.9
7.10
7.11
Copulas
Application to loan portfolios
Summary
Further reading
Questions and problems
Assignment questions
152
159
161
161
162
163
regulation and Basel II
Reasons for regulating bank capital
Pre-1988
The 1988 BIS Accord
The G-30 policy recommendations
Netting
The 1996 Amendment
Basel II
Credit risk capital under Basel II
Operational risk under Basel II
Supervisory review
Market discipline
Summary
Further reading
Questions and problems
Assignment questions
165
167
168
169
172
174
176
178
179
188
189
190
191
192
192
194
Chapter 8 The VaR measure
195
8.1
Definition of VaR
196
8.2
VaR vs. expected shortfall
198
8.3
Properties of risk measures
199
8.4
Choice of parameters for VaR
202
8.5
Marginal VaR, incremental VaR, and component VaR... 206
8.6
Back testing
208
8.7
Stress testing
212
Summary
213
Further reading
214
Questions and problems
215
Assignment questions
216
Chapter 9 Market risk VaR: historical simulation approach
9.1
The methodology
9.2
Accuracy
9.3
Extensions
9.4
Extreme value theory
9.5
Application
Summary
Further reading
217
217
220
221
224
227
229
230
x
Contents
Questions and problems
Assignment questions
231
231
Chapter 10 Market risk VaR: model-building approach
10.1 The basic methodology
10.2 The linear model
10.3 Handling interest rates
10.4 Applications of the linear model
10.5 The linear model and options
10.6 The quadratic model
10.7 Monte Carlo simulation
10.8 Using distributions that are not normal
10.9 Model building vs. historical simulation
Summary
Further reading
Questions and problems
Assignment questions
233
234
237
238
242
242
246
248
249
250
251
252
252
254
Chapter 11 Credit risk: estimating default probabilities
11.1 Credit ratings
11.2 Historical default probabilities
11.3 Recovery rates
11.4 Estimating default probabilities from bond prices
11.5 Comparison of default probability estimates
11.6 Using equity prices to estimate default probabilities
Summary
Further reading
Questions and problems
Assignment questions
255
255
258
260
261
265
269
272
273
273
275
Chapter 12 Credit risk losses and credit VaR
12.1 Estimating credit losses
12.2 Credit risk mitigation
12.3 Credit VaR
12.4 Vasicek's model
12.5 Credit Risk Plus
12.6 CreditMetrics
12.7 Interpreting credit correlations
Summary
Further reading
Questions and problems
Assignment questions
277
278
283
287
287
288
289
293
295
295
296
297
Chapter 13 Credit derivatives
13.1 Credit default swaps
13.2 Credit indices
299
299
303
xi
Contents
13.3
13.4
13.5
13.6
13.7
13.8
Valuation of credit default swaps
CDS forwards and options
Total return swaps
Basket credit default swaps
Collateralized debt obligations
Valuation of a basket CDS and CDO
Summary
Further reading
Questions and problems
Assignment questions
303
308
310
311
311
314
316
317
317
319
Chapter 14 Operational risk
14.1 What is operational risk?
14.2 Determination of regulatory capital
14.3 Categorization of operational risks
14.4 Loss severity and loss frequency
14.5 Forward looking approaches
14.6 Allocation of operational risk capital
14.7 Use of the power law
14.8 Insurance
14.9 Sarbanes-Oxley
Summary
Further reading
Questions and problems
Assignment questions
321
323
324
326
327
331
333
335
335
337
338
339
340
340
Chapter 15 Model risk and liquidity risk
15.1 The nature of models in finance
15.2 Models for nonlinear products
15.3 Models for actively traded products
15.4 Models for structured products
15.5 Dangers in model building
15.6 Detecting model problems
15.7 Traditional view of liquidity risk
15.8 Liquidity black holes
15.9 Long-term capital management
15.10 Liquidity vs. profitability
Summary
Further reading
Questions and problems
Assignment questions
343
344
345
346
351
352
353
354
356
360
361
362
363
363
364
Chapter 16 Economic capital and RAROC
16.1 Definition of economic capital
16.2 Components of economic capital
365
366
368
xii
Contents
16.3
16.4
16.5
16.6
16.7
16.8
Shapes of the loss distributions
Relative importance of risks
Aggregating economic capital
Allocation of the diversification benefit
Deutsche Bank's economic capital
RAROC
Summary
Further reading
Questions and problems
Assignment questions
Chapter 17 Weather, energy, and insurance derivatives
17.1 Weather derivatives
17.2 Energy derivatives
17.3 Insurance derivatives
Summary
Further reading
Questions and problems
Assignment question
Chapter 18 Big losses and what we can learn from them
18.1 Risk limits
18.2 Managing the trading room
18.3 Liquidity risk
18.4 Lessons for nonfinancial corporations
Summary
Further reading
Appendix A: Valuing forward and futures contracts
370
372
373
377
378
379
381
381
381
382
385
385
387
390
391
392
393
394
395
397
399
402
404
406
406
407
Appendix B: Valuing swaps
Appendix C: Valuing European options
409
413
Appendix D: Valuing American options
Appendix E: The manipulation of credit transition matrices
417
421
Answers to questions and problems
Glossary of terms
423
457
DerivaGem software
Table for N(x) when x < 0
479
484
Table for N(x) when x > 0
Index
485
487
BUSINESS SNAPSHOTS
1.1
1.2
2.1
2.2
2.3
2.4
3.1
3.2
3.3
4.1
5.1
5.2
7.1
7.2
8.1
11.1
12.1
12.2
13.1
13.2
13.3
14.1
14.2
15.1
15.2
15.3
15.4
15.5
15.6
16.1
18.1
The Hidden Costs of Bankruptcy
Expensive Failures of Financial Institutions in the US
The Unanticipated Delivery of a Futures Contract
Procter and Gamble's Bizarre Deal
Microsoft's Hedging
The Barings Bank Disaster
Hedging by Gold Mining Companies
Dynamic Hedging in Practice
Is Delta Hedging Easier or More Difficult for Exotics
Orange County's Yield Curve Plays
What Causes Volatility?
Making Money from Foreign Currency Options
Systemic Risk
Basel III?
Historical Perspectives on VaR
Risk-Neutral Valuation
Long-Term Capital Management's Big Loss
Downgrade Triggers and Enron's Bankruptcy
Who Bears the Credit Risk?
Is the CDS Market a Fair Game?
Correlation Smiles
The Hammersmith and Fulham Story
Rogue Trader Insurance
Kidder Peabody's Embarrassing Mistake
Exploiting the Weaknesses of a Competitor's Model
Crashophobia
The Crash of 1987
British Insurance Companies
Metallgesellschaft
The EGT Fund
Big Losses
14
22
35
39
47
49
58
72
73
85
113
118
168
189
196
267
285
286
301
309
315
332
337
345
346
349
358
359
362
373
396
Preface
This book is based on an elective course entitled Financial Risk Management I have taught at University of Toronto for many years. The main
focus of the book is on the risks faced by banks and other financial
institutions, but much of the material presented is equally important to
nonfinancial institutions. Like my popular text Options, Futures, and
Other Derivatives, this book is designed to be useful to practitioners as
well as college students.
The book is appropriate for elective courses in either risk management
or the management of financial institutions. It is not necessary for students
to take a course on options and futures markets prior to taking a course
based on this book, but if they have taken such a course much of the
material in the first four chapters will not need to be covered. Chapter 13
on credit derivatives and Chapter 17 on weather, energy, and insurance
derivatives can be skipped if this material is covered elsewhere or is not
considered appropriate. Chapter 18 on big losses and what we can learn
from them is a great chapter for the last class of a course because it draws
together many of the points made in earlier chapters.
The level of mathematical sophistication in the way material is presented has been managed carefully so that the book is accessible to as
wide an audience as possible. For example, when covering copulas in
Chapter 6, I present the intuition followed by a detailed numerical
example; when covering maximum-likelihood methods in Chapter 5
and extreme value theory in Chapter 9, I provide numerical examples
and enough details for readers to develop their own Excel spreadsheets.
This is a book about risk management and so there is very relatively little
material on the valuation of derivatives contracts. (This is the main focus
my other two books Options, Futures, and Other Derivatives and
Fundamentals of Futures and Options Markets.) For reference, I have
included at the end of the book appendices that summarize some of
the key derivatives pricing formulas that are important to risk managers.
xvi
Preface
Slides
Several hundred PowerPoint slides can be downloaded from my website.
Instructors who adopt the text are welcome to adapt the slides to meet
their own needs.
Questions and Problems
End-of-chapter problems are divided into two groups: "Questions and
Problems" and "Assignment Questions". Solutions to Questions and
Problems are at the end of the book, while solutions to Assignment
Questions are made available by the publishers to adopting instructors
in the Instructors' Manual.
Acknowledgments
Many people have played a part in the production of this book. I have
benefited from interactions with many academics and risk managers. I
would like to thank the students in my MBA Financial Risk Management
elective course who have made many suggestions as to how successive
drafts of the material could be improved. I am particularly grateful to
Ateet Agarwal, Ashok Rao, and Yoshit Rastogi who provided valuable
research assistance as the book neared completion. Eddie Mizzi from
The Geometric Press did an excellent job editing the final manuscript and
handling the page composition.
Alan White, a colleague at the University of Toronto, deserves a special
acknowledgment. Alan and I have been carrying out joint research and
consulting in the area of derivatives and risk management for over twenty
years. During that time we have spent countless hours discussing key
issues. Many of the new ideas in this book, and many of the new ways
used to explain old ideas, are as much Alan's as mine.
Special thanks are due to many people at Prentice Hall, particularly my
editor David Alexander, for their enthusiasm, advice, and encouragement.
I welcome comments on the book from readers. My e-mail address is:
hull@rotman.utoronto.ca
John Hull
Joseph L. Rotman School of Management
University of Toronto
Introduction
Imagine you are the Chief Risk Officer of a major corporation. The CEO
wants your views on a major new venture. You have been inundated with
reports showing that the new venture has a positive net present value and
will enhance shareholder value. What sort of analysis and ideas is the
CEO looking for from you?
As Chief Risk Officer it is your job to consider how the new venture fits
into the company's portfolio. What is the correlation of the performance
of the new venture with the rest of the company's business? When the rest
of the business is experiencing difficulties, will the new venture also
provide poor returns—or will it have the effect of dampening the ups
and downs in the rest of the business?
Companies must take risks if they are to survive and prosper. The risk
management function's primary responsibility is to understand the portfolio of risks that the company is currently taking and the risks it plans to
take in the future. It must decide whether the risks are acceptable and, if
they are not acceptable, what action should be taken.
Most of this book is concerned with the ways risks are managed by
banks and other financial institutions, but many of the ideas and
approaches we will discuss are equally applicable to other types of
corporations. Risk management is now recognized as a key activity
for all corporations. Many of the disastrous losses of the 1990s, such
as those at Orange County in 1994 and Barings Bank in 1995, would
have been avoided if good risk management practices had been in place.
2
Chapter
This opening chapter sets the scene. It starts by reviewing the classica
arguments concerning the risk/return trade-offs faced by an investor who
is choosing a portfolio of stocks and bonds. It then considers whether the
same arguments can be used by a company in choosing new projects and
managing its risk exposure. After that the focus shifts to banks. The
chapter looks at a typical balance sheet and income statement for a bank
and examines the key role of capital in cushioning the bank from adverse
events. It takes a first look at the main approaches used by a bank in
managing its risks and explains how a bank avoids fluctuations in net
interest income.
1.1 RISK vs. RETURN FOR INVESTORS
As all fund managers know, there is a trade-off between risk and return
when money is invested. The greater the risks taken, the higher the return
that can be realized. The trade-off is actually between risk and expected
return, not between risk and actual return. The term "expected return'
sometimes causes confusion. In everyday language an outcome that is
"expected" is considered likely to occur. However, statisticians define the
expected value of a variable as its mean value. Expected return is therefore
a weighted average of the possible returns where the weight applied to a
particular return equals the probability of that return occurring.
Suppose, for example, that you have $100,000 to invest for one year
One alternative is to buy Treasury bills yielding 5% per annum. There if
then no risk and the expected return is 5%. Another alternative is to
invest the $100,000 in a stock. To simplify things a little, we suppose that
the possible outcomes from this investment are as shown in Table 1.1
There is a 0.05 probability that the return will be +50%; there is a 0.2'
probability that the return will be +30%; and so on. Expressing the
Table 1.1 Return in one year from
investing $100,000 in equities.
Probability
Return
0.05
0.25
0.40
0.25
0.05
+50%
+30%
+10%
-10%
-30%
Introduction
3
returns in decimal form, the expected return per year is
0 05 x 0.50 + 0.25 x 0.30 + 0.40 x 0.10
+ 0.25 x (-0.10) + 0.05 x (-0.30) = 0.10
This shows that in return for taking some risk you are able to increase your
expected return per annum from the 5% offered by Treasury bills to 10%.
If things work out well, your return per annum could be as high as 50%.
However, the worst-case outcome is a —30% return, or a loss of $30,000.
One of the first attempts to understand the trade-off between risk and
expected return was by Markowitz (1952). Later Sharpe (1964) and others
carried the Markowitz analysis a stage further by developing what is
known as the capital asset pricing model. This is a relationship between
expected return and what is termed systematic risk. In 1976 Ross developed
arbitrage pricing theory—an extension of the capital asset pricing model
to the situation where there are several sources of systematic risk. The key
insights of these researchers have had a profound effect on the way
portfolio managers think about and analyze the risk/return trade-offs
that they face. In this section we review these insights.
Quantifying Risk
How do you quantify the risk you take when choosing an investment?
A convenient measure that is often used is the standard deviation of
return over one year. This is
where R is the return per annum. The symbol E denotes expected value, so
that E(R) is expected return per annum. In Table 1.1, as we have shown,
E(R) = 0.10. To calculate E(R2), we must weight the alternative squared
returns by their probabilities:
E(R2) - 0.05 x 0.502 + 0.25 x 0.302 + 0.40 x 0.102
+ 0.25 x (-0.10) 2 + 0.05 x (-0.30) 2 = 0.046
The standard deviation of returns is therefore V 0 . 0 4 6 - 0 . 1 2 = 0.1897,
or 18.97%.
Investment Opportunities
Suppose we choose to characterize every investment opportunity by its
expected return and standard deviation of return. We can plot available
4
Chapter 1
Figure 1.1
Alternative risky investments.
risky investments on a chart such as Figure 1.1, where the horizontal axis
is the standard deviation of return and the vertical axis is the expected
return.
Once we have identified the expected return and the standard deviation
of return for individual investments, it is natural to think about what
happens when we combine investments to form a portfolio. Consider two
investments with returns R1 and R2. The return from putting a proportion
of your money in the first investment and a proportion
in the second investment is
The expected return of the portfolio is
(1.1)
where
is the expected return from the first investment and
is the
expected return from the second investment. The standard deviation of
the portfolio return is given by
(1.2)
Introduction
5
Table 1.2 Expected return,
and standard deviation of return,
from a portfolio consisting of two investments. The expected
returns from the investments are 10% and 15%, the standard
deviation of the returns are 16% and 24%, and the correlation
between the returns is 0.2.
0.0
0.2
0.4
0.6
0.8
1.0
1.0
0.8
0.6
0.4
0.2
0.0
15%
14%
13%
12%
11%
10%
24.00%
20.09%
16.89%
14.87%
14.54%
16.00%
where
and
are the standard deviations of R1 and R2, and is the
coefficient of correlation between the two.
Suppose that
is 10% per annum and
is 16% per annum, while
is 15% per annum and
is 24% per annum. Suppose also that the
coefficient of correlation,
between the returns is 0.2 or 20%. Table 1.2
Figure 1.2 Alternative risk/return combinations from two investments
as calculated in Table 1.2.
6
Chapter 1
shows the values of
and
for a number of different values of
and
The calculations show that by putting part of your money in the first
investment and part in the second investment a wide range of risk/return
combinations can be achieved. These are plotted in Figure 1.2.
Most investors are risk-averse. They want to increase expected return
while reducing the standard deviation of return. This means that they
want to move as far as they can in a "north-west" direction in Figures 1.1
and 1.2. As we saw in Figure 1.2, forming a portfolio of the two
investments that we considered helps them do this. For example, by
putting 60% in the first investment and 40% in the second, a portfolio
with an expected return of 12% and a standard deviation of return equal
to 14.87% is obtained. This is an improvement over the risk/return tradeoff for the first investment. (The expected return is 2% higher and the
standard deviation of the return is 1.13% lower.)
Efficient Frontier
Let us now bring a third investment into our analysis. The third investment can be combined with any combination of the first two investments
to produce new risk/return trade-offs. This enables us to move further in
the north-west direction. We can then add a fourth investment. This can
be combined with any combination of the first three investments to
produce yet more investment opportunities. As we continue this process,
Figure 1.3
The efficient frontier of risky investments.
Introduction
7
considering every possible portfolio of the available risky investments in
Figure 1.1, we obtain what is known as an efficient frontier. This represents the limit of how far we can move in a north-west direction and is
illustrated in Figure 1.3. There is no investment that dominates a point on
the efficient frontier in the sense that it has both a higher expected return
and a lower standard deviation of return. The shaded area in Figure 1.3
represents the set of all investments that are possible. For any point in the
shaded area, we can find a point on the efficient frontier that has a better
(or equally good) risk/return trade-off.
In Figure 1.3 we have considered only risky investments. What does the
efficient frontier of all possible investments look like? To consider this, we
first note that one available investment is the risk-free investment. Suppose that the risk-free investment yields a return of RF. In Figure 1.4 we
have denoted the risk-free investment by point F and drawn a tangent
Figure 1.4 The efficient frontier of all investments. Point I is achieved by
investing a percentage of available funds in portfolio M and the rest in a
risk-free investment. Point J is achieved by borrowing
- 1 of available
funds at the risk-free rate and investing everything in portfolio M.
from point F to the efficient frontier of risky investments. M is the point
of tangency. As we will now show, the line FM is our new efficient
frontier.
Consider what happens when we form an investment I by putting
(0 <
< 1) of the funds we have available for investment in the risky
portfolio, M, and 1 n the risk-free investment, F. From equation (1.1)
the expected return from the investment, E{RI), is given by
E(RI) = {1-
)RF+
E{RM)
and from equation (1.2) the standard deviation of this return is
,
where
is the standard deviation of returns for portfolio M. This
risk/return combination corresponds to point labeled I in Figure 1.4.
From the perspective of both expected return and standard deviation of
return, point I is
of the way from F to M.
All points on the line FM can be obtained by choosing a suitable
combination of the investment represented by point F and the investment
represented by point M. The points on this line dominate all the points on
the previous efficient frontier because they give a better risk/return tradeoff. The straight line FM is therefore the new efficient frontier.
If we make the simplifying assumption that we can borrow at the riskfree rate of RF as well as invest at that rate, we can create investments that
are on the line from F to M but beyond M. Suppose, for example, that we
want to create the investment represented by the point J in Figure 1.4,
where the distance of J from F is
( > 1) times the distance of M
from F. We borrow
- 1 of the amount that we have available for
investment at rate RF and then invest everything (the original funds and
the borrowed funds) in the investment represented by point M. After
allowing for the interest paid, the new investment has an expected return,
E(Rj), given by
E(Rj) =
E(RM) - ( - 1)RF
and the standard deviation of the return is
, This shows that the risk
and expected return combination corresponds to point J.
The argument that we have presented shows that, when the risk-free
investment is considered, the efficient frontier must be a straight line. To
put this another way, there should be a linear trade-off between the
expected return and the standard deviation of returns, as indicated in
Figure 1.4. All investors should choose the same portfolio of risky assets.
This is the portfolio represented by M. They should then reflect their
9
Introduction
appetite for risk by combining this risky investment with borrowing or
lending at the risk-free rate.
It is a short step from here to argue that the portfolio of risky
investments represented by M must be the portfolio of all risky investments. How else could it be possible that all investors hold the portfolio?
The amount of a particular risky investment in portfolio M must be
proportional to the amount of that investment available in the economy.
The investment M is usually referred to as the market portfolio.
Systematic vs. Nonsystematic Risk
How do investors decide on the expected returns they require for individual investments? Based on the analysis we have presented, the market
portfolio should play a key role. The expected return required on an
investment should reflect the extent to which the investment contributes
to the risks of the market portfolio.
A common procedure is to use historical data to determine a best-fit
linear relationship between returns from an investment and returns from
the market portfolio. This relationship has the form
R=
(1.3)
where R is the return from the investment, RM is the return from the
market portfolio, and are constants, and is a random variable equal
to the regression error.
Equation (1.3) shows that there are two components to the risk in the
investment's return:
1. A component RM, which is a multiple of the return from the
market portfolio
2. A component
which is unrelated to the return from the market
portfolio
The first component is referred to as systematic risk; the second component is referred to as nonsystematic risk.
Consider first the nonsystematic risk. If we assume that the
for
different investments are independent of each other, the nonsystematic risk
is almost completely diversified away in a large portfolio. An investor
should not therefore be concerned about nonsystematic risk and should
not require an extra return above the risk-free rate for bearing nonsystematic risk.
The systematic risk component is what should matter to an investor.
Chapter 1
10
When a large well-diversified portfolio is held, the systematic risk represented by RM does not disappear. An investor should require an
expected return to compensate for this systematic risk.
We know how investors trade off systematic risk and expected return
from Figure 1.4. When = 0, there is no systematic risk and the expected
return is RF. When = 1, we have the same systematic risk as point M
and the expected return should be E{RM). In general,
E(R) = RF+ [E(RM)-RF]
(1.4)
This is the capital asset pricing model. The excess expected return over the
risk-free rate required on the investment is times the excess expected
return on the market portfolio. This relationship is plotted in Figure 1.5.
Suppose that the risk-free rate is 5% and the return on the market portfolio
is 10%. An investment with a of 0 should have an expected return of 5%;
an investment with a of 0.5 should have an expected return of 7.5%; an
investment with a
of 1.0 should have an expected return of 10%;
and so on.
The variable is the beta of the investment. It is equal to
where
is the correlation between the return from the investment and the return
from the market portfolio, is the standard deviation of the return from
the investment, and
is the standard deviation of the return from the
Figure 1.5 The capital asset pricing model.
Introduction
11
market portfolio. Beta measures the sensitivity of the return from the
investment to the return from the market portfolio. We can define the
beta of any investment portfolio in a similar way to equation (1.3). The
capital asset pricing model in equation (1.4) should then apply with the
return R defined as the return from the portfolio.
In Figure 1.4 the market portfolio represented by M has a beta of 1.0
and the riskless portfolio represented by F has a beta of zero. The
portfolios represented by the points I and J have betas equal to
and
respectively.
Assumptions
The analysis we have presented makes a number of simplifying assumptions. In particular, it assumes:
1. Investors care only about the expected return and the standard
deviation of return
2. The
for different investments in equation (1.3) are independent
3. Investors focus on returns over just one period and the length of this
period is the same for all investors
4. Investors can borrow and lend at the same risk-free rate
5. There are no tax considerations
6. All investors make the same estimates of expected returns, standard
deviations of returns, and correlations for available investments
These assumptions are of course not exactly true. Investors have complex
sets of risk preferences that involve more than just the first two moments
of the return distribution. The one-factor model in equation (1.3) assumes
that the correlation between the returns from two investments arises only
from their correlations with the market portfolio. This is clearly not true
for two investments in the same sector of the economy. Investors have
different time horizons. They cannot borrow and lend at the same rate.
Taxes do influence the portfolios that investors choose and investors do
not have homogeneous expectations. (Indeed, if the assumptions of the
capital asset pricing model held exactly, there would be very little trading.)
In spite of all this, the capital asset pricing model has proved to be a
useful tool for portfolio managers. Estimates of the betas of stocks are
readily available and the expected return on a portfolio estimated by the
capital asset pricing model is a commonly used benchmark for assessing
the performance of the portfolio manager.
12
Chapter 1
Arbitrage Pricing Theory
A more general analysis that moves us away from the first two assumptions listed above is arbitrage pricing theory. The return from an investment is assumed to depend on several factors. By exploring ways in which
investors can form portfolios that eliminate their exposure to the factors,
arbitrage pricing theory shows that the expected return from an investment is linearly dependent on the factors.
The assumption that the
for different investments are independent
in equation (1.3) ensures that there is just one factor driving expected
returns (and therefore one source of systematic risk) in the capital asset
pricing model. This is the return from the market portfolio. In arbitrage
pricing theory there are several factors affecting investment returns. Each
factor is a separate source of systematic risk. Unsystematic risk is the risk
that is unrelated to all the factors and can be diversified away.
1.2 RISK vs. RETURN FOR COMPANIES
We now move on to consider the trade-offs between risk and return
made by a company. How should a company decide whether the
expected return on a new investment project is sufficient compensation
for its risks?
The ultimate owners of a company are its shareholders and a company should be managed in the best interests of its shareholders. It is
therefore natural to argue that a new project undertaken by the company should be viewed as an addition to its shareholder's portfolio. The
company should calculate the beta of the investment project and its
expected return. If the expected return is greater than that given by the
capital asset pricing model, it is a good deal for shareholders and the
investment should be accepted. Otherwise it should be rejected. The
argument suggests that nonsystematic risks should not be considered
in the accept/reject decision.
In practice, companies are concerned about nonsystematic as well as
systematic risks. For example, most companies insure themselves against
the risk of their buildings being burned down—even though this risk is
entirely nonsystematic and can be diversified away by their shareholders.
They try to avoid taking high risks and often hedge their exposures to
exchange rates, interest rates, commodity prices, and other market variables. Earnings stability and the survival of the company are important
managerial objectives. Companies do try and ensure that the expected
Introduction
13
returns on new ventures are consistent with the risk/return trade-offs of
their shareholders. But there is an overriding constraint that the risks
taken should not be allowed to get too large.
Most investors are also concerned about the overall risk of the companies they invest in. They do not like surprises and prefer to invest in
companies that show solid growth and meet earnings forecasts. They like
companies to manage risks carefully and limit the overall amount of
risk—both systematic and nonsystematic—they are taking.
The theoretical arguments we presented in Section 1.1 suggest that
investors should not behave in this way. They should encourage companies
to make high-risk investments when the trade-off between expected return
and systematic risk is favorable. Some of the companies in a shareholder's
portfolio will go bankrupt, but others will do very well. The result should
be an overall return to the shareholder that is satisfactory.
Are investors behaving suboptimally? Would their interests be better
served if companies took more nonsystematic risks because investors are
in a position to diversify away these risks? There is an important argument to suggest that this is not necessarily the case. This argument is
usually referred to as the "bankruptcy costs" argument. It is often used to
explain why a company should restrict the amount of debt it takes on, but
it can be extended to apply to all risks.
Bankruptcy Costs
In a perfect world, bankruptcy would be a fast affair where the company's
assets (tangible and intangible) are sold at their fair market value and the
proceeds distributed to bondholders, shareholders, and other stakeholders
using well-defined rules. If we lived in such a perfect world, the bankruptcy
process itself would not destroy value for shareholders. Unfortunately the
real world is far from perfect. The bankruptcy process leads to what are
known as bankruptcy costs.
What is the nature of bankruptcy costs? Once a bankruptcy has been
announced, customers and suppliers become less inclined to do business
with the company; assets sometimes have to be sold quickly at prices well
below those that would be realized in an orderly sale; the value of
important intangible assets such as the company's brand name and its
reputation in the market are often destroyed; the company is no longer
run in the best interests of shareholders; large fees are often paid to
accountants and lawyers; and so on. Business Snapshot 1.1 is a fictitious
story, but all too representative of what happens in practice. It illustrates
Chapter 1
14
Business Snapshot 1.1
The Hidden Costs of Bankruptcy
Several years ago a company had a market capitalization of $2 billion and
$500 million of debt. The CEO decided to acquire a company in a related
industry for $1 billion in cash. The cash was raised using a mixture of bank debt
and bond issues. The price paid for the company was close to its market value
and therefore presumably reflected the market's assessment of the company's
expected return and its systematic risk at the time of the acquisition.
Many of the anticipated synergies used to justify the acquisition were not
realized. Furthermore the company that was acquired was not profitable. After
three years the CEO resigned. The new CEO sold the acquisition for $100 million (10% of the price paid) and announced the company would focus on its
original core business. However, by then the company was highly levered. A
temporary economic downturn made it impossible for the company to service
its debt and it declared bankruptcy.
The offices of the company were soon filled with accountants and lawyers
representing the interests of the various parties (banks, different categories of
bondholders, equity holders, the company, and the board of directors).
These people directly or indirectly billed the company about $10 million
per month in fees. The company lost sales that it would normally have made
because nobody wanted to do business with a bankrupt company. Key senior
executives left. The company experienced a dramatic reduction in its market
share.
After two years and three reorganization attempts, an agreement was
reached between the various parties and a new company with a market
capitalization of $700,000 was incorporated to continue the remaining profitable parts of the business. The shares in the new company were entirely owned
by the banks and the bondholders. The shareholders got nothing.
how, when a high-risk decision works out badly, there can be disastrous
bankruptcy costs.
We mentioned earlier that corporate survival is an important managerial objective and that shareholders like companies to avoid excessive risks.
We now understand why this is so. Bankruptcy laws vary widely from
country to country, but they all have the effect of destroying value as
lenders and other creditors vie with each other to get paid. This value has
often been painstakingly built up by the company over many years. It
makes sense for a company that is operating in the best interests of its
shareholders to limit the probability of this value destruction occurring. It
does this by limiting the total risk (systematic and nonsystematic) that
it takes.
15
Introduction
When a major new investment is being contemplated, it is important to
consider how well it fits in with other risks taken by the company.
Relatively small investments can often have the effect of reducing the
overall risks taken by a company. However, a large investment can
dramatically increase these risks. Many spectacular corporate failures
(such as the one in Business Snapshot 1.1) can be traced to CEOs who
made large acquisitions (often highly levered) that did not work out.
1.3 BANK CAPITAL
We now switch our attention to banks. Banks face the same types of
bankruptcy costs as other companies and have an incentive to manage
their risks (systematic and nonsystematic) prudently so that the probability of bankruptcy is minimal. Indeed, as we shall see in Chapter 7,
governments regulate banks in an attempt to ensure that they do exactly
this. In this section we take a first look at the nature of the risks faced by a
bank and discuss the amount of capital banks need.
Consider a hypothetical bank DLC (Deposits and Loans Corporation).
DLC is primarily engaged in the traditional banking activities of taking
deposits and making loans. A summary balance sheet for DLC at the end
of 2006 is shown in Table 1.3 and a summary income statement for 2006
is shown in Table 1.4.
Table 1.3 shows that the bank has $100 billion of assets. Most of the
assets (80% of the total) are loans made by the bank to private individuals
and corporations. Cash and marketable securities account for a further
15% of the assets. The remaining 5% of the assets are fixed assets (i.e.,
buildings, equipment, etc.). A total of 90% of the funding for the assets
comes from deposits of one sort or another from the bank's customers and
counterparties. A further 5% is financed by subordinated long-term debt
Table 1.3 Summary balance sheet for DLC at end of 2006 ($ billions).
Liabilities and net worth
Assets
Cash
Marketable securities
Loans
Fixed assets
Total
5
10
80
5
100
Deposits
Subordinated long-term debt
Equity capital
Total
90
5
5
100
Chapter 1
16
Table 1.4 Summary income statement for
DLC in 2006 ($ billions).
Net interest income
Loan losses
Noninterest income
Noninterest expense
Pre-tax operating income
3.00
(0.80)
0.90
(2.50)
0.60
(i.e., bonds issued by the bank to investors that rank below deposits in the
event of a liquidation) and the remaining 5% is financed by the bank's
shareholders in the form of equity capital. The equity capital consists of
the original cash investment of the shareholders and earnings retained in
the bank.
Consider next the income statement for 2006 shown in Table 1.4. The
first item on the income statement is net interest income. This is the excess
of the interest earned over the interest paid and is 3% of assets in our
example. It is important for the bank to be managed so that net interest
income remains roughly constant regardless of the level of interest rates.
We will discuss this in more detail in Section 1.5.
The next item is loan losses. This is 0.8% of assets for the year in
question. Even if a bank maintains a tight control over its lending policies,
this will tend to fluctuate from year to year. In some years default rates in
the economy are high; in others they are quite low.1 The management and
quantification of the credit risks it takes is clearly of critical importance to
a bank. This will be discussed in Chapters 11, 12, and 13.
The next item, noninterest income, consists of income from all the
activities of a bank other than lending money. This includes trading
activities, fees for arranging debt or equity financing for corporations,
and fees for the many other services the bank provides for its retail and
corporate clients. In the case of DLC, noninterest income is 0.9% of
assets. This must be managed carefully. In particular, the market risks
associated with trading activities must be quantified and controlled.
Market risk management procedures are discussed in Chapters 3, 8, 9,
and 10.
The final item is noninterest expense and is 2.5% of assets in our
1
Evidence for this can be found by looking at Moody's statistics on the default rates on
bonds between 1970 and 2003. This ranged from a low of 0.09% in 1979 to a high of
3.81% in 2001.
17
Introduction
example. This consists of all expenses other than interest paid. It includes
salaries, technology-related costs, and other overheads. As in the case of
all large businesses, these have a tendency to increase over time unless
they are managed carefully. Banks must try and avoid large losses from
litigation, business disruption, employee fraud, etc. The risk associated
with these types of losses is known as operational risk and will be
discussed in Chapter 14.
Capital Adequacy
Is the equity capital of 5% of assets in Table 1.3 adequate? One way of
answering this is to consider an extreme scenario and determine whether
the bank will survive. Suppose that there is a severe recession and as a
result the bank's loan losses rise to 4% next year. We assume that other
items on the income statement are unaffected. The result will be a pre-tax
net operating loss of 2.6% of assets. Assuming a tax rate of 30%, this
would result in an after-tax loss of about 1.8% of assets.
In Table 1.3 equity capital is 5% of assets and so an after-tax loss equal
to 1.8% of assets, although not at all welcome, can be absorbed. It would
result in a reduction of the equity capital to 3.2% of assets. Even a second
bad year similar to the first would not totally wipe out the equity.
Suppose now that the bank has the more aggressive capital structure
shown in Table 1.5. Everything is the same as Table 1.3 except that equity
capital is 1% (rather than 5%) of assets and deposits are 94% (rather
than 90%) of assets. In this case one year where the loan losses are 4% of
assets would totally wipe out equity capital and the bank would find itself
in serious financial difficulties. It would no doubt try to raise additional
equity capital, but it is likely to find this almost impossible in such a weak
financial position.
Table 1.5
Alternative balance sheet for DLC at end of 2006 with
equity only 1 % of assets (billions).
Assets
Cash
Marketable securities
Loans
Fixed assets
Total
Liabilities and net worth
5
10
80
5
100
Deposits
Subordinated long-term debt
Equity capital
Total
94
5
1
100
18
Chapter 1
It is quite likely that there would be a run on deposits and the bank
would be forced into liquidation. If all assets could be liquidated for book
value (a big assumption), the long-term debtholders would likely receive
about $4.2 billion rather than $5 billion (they would in effect absorb the
negative equity) and the depositors would be repaid in full.
Note that equity and subordinated long-term debt are both sources of
capital. Equity provides the best protection against adverse events. (In
our example, when the bank has $5 billion of equity capital rather than
$1 billion, it stays solvent and is unlikely to be liquidated.) Subordinated
long-term debt holders rank below depositors in the event of a default.
However, subordinated long-term debt does not provide as good a
cushion for the bank as equity. As our example shows, it does not
necessarily prevent the bank's liquidation.
As we shall see in Chapter 7, bank regulators have been active in
ensuring that the capital a bank keeps is sufficient to cover the risks it
takes. Regulators consider the market risks from trading activities as well
as the credit risks from lending activities. They are moving toward an
explicit consideration of operational risks. Regulators define different
types of capital and prescribe levels for them. In our example DLC's
equity capital is Tier 1 capital; subordinated long-term debt is Tier 2
capital.
1.4 APPROACHES TO MANAGING RISKS
Since a bank's equity capital is typically very low in relation to the assets
on the balance sheet, a bank must manage its affairs conservatively to
avoid large fluctuations in its earnings. There are two broad risk management strategies open to the bank (or any other organization). One
approach is to identify risks one by one and handle each one separately.
This is sometimes referred to as risk decomposition. The other is to reduce
risks by being well diversified. This is sometimes referred to as risk
aggregation. In practice, banks use both approaches when they manage
market and credit risks as we will now explain.
Market Risks
Market risks arise primarily from the bank's trading activities. A bank
has exposure to interest rates, exchange rates, equity prices, commodity
prices, and the market variables. These risks are in the first instance
managed by the traders. For example, there is likely to be one trader (or a
Introduction
19
group of traders) working for a US bank who is responsible for the
dollar/yen exchange rate risk. At the end of each day the trader is
required to ensure that risk limits specified by the bank are not exceeded.
If the end of the day is approached and one or more of the risk limits is
exceeded, the trader must execute new hedging trades so that the limits
are adhered to.
The risk managers working for the bank then aggregate the residual
market risks from the activities of all traders to determine the total risk
faced by the bank from movements in market variables. Hopefully the
bank is well diversified, so that its overall exposure to market movements
is fairly small. If risks are unacceptably high, the reasons must be
determined and corrective action taken.
Credit Risks
Credit risks are traditionally managed by ensuring that the credit portfolio is well diversified (risk aggregation). If a bank lends all its available
funds to a single borrower, then it is totally undiversified and subject to
huge risks. If the borrowing entity runs into financial difficulties and is
unable to make its interest and principal payments, the bank will
become insolvent.
If the bank adopts a more diversified strategy of lending 0.01% of its
available funds to each of 10,000 different borrowers, then it is in a
much safer position. Suppose that in a typical year the probability of
any one borrower defaulting is 1%. We can expect that close to 100
borrowers will default in the year and the losses on these borrowers will
be more than offset by the profits earned on the 99% of loans that
perform well.
Diversification reduces nonsystematic risk. It does not eliminate systematic risk. The bank faces the risk that there will be an economic
downturn and a resulting increase in the probability of default by
borrowers. To maximize the benefits of diversification, borrowers should
be in different geographical regions and different industries. A large
international bank with different types of borrowers all over the world
is likely to be much better diversified than a small bank in Texas that
lends entirely to oil companies. However, there will always be some
systematic risks that cannot be diversified away. For example, diversification does not protect a bank against world economic downturns.
Since the late 1990s we have seen the emergence of an active market for
credit derivatives. Credit derivatives allow banks to handle credit risks
Chapter 1
20
one by one (risk decomposition) rather than relying solely on risk
diversification. They also allow banks to buy protection against the
overall level of defaults in the economy. We discuss credit derivatives in
Chapter 13.
1.5 THE MANAGEMENT OF NET INTEREST INCOME
As mentioned earlier net interest income is the excess of interest received
over interest paid. It is the role of the asset/liability management function
to ensure that fluctuations in net interest income are minimal. In this
section we explain how it does this.
To illustrate how fluctuations in net interest income could occur,
consider a simple situation where a bank offers consumers a one-year
and a five-year deposit rate as well as a one-year and five-year mortgage
rate. The rates are shown in Table 1.6. We make the simplifying assumption that market participants expect the one-year interest rate for future
time periods to equal the one-year rates prevailing in the market today.
Loosely speaking, this means that the market considers interest rate
increases to be just as likely as interest rate decreases. As a result the rates
in Table 1.6 are "fair", in that they reflect the market's expectations.
Investing money for one year and reinvesting for four further one-year
periods gives the same expected overall return as a single five-year
investment. Similarly, borrowing money for one year and refinancing each
year for the next four years leads to the same expected financing costs as a
single five-year loan.
Now suppose you have money to deposit and agree with the prevailing
view that interest rate increases are just as likely as interest rate decreases.
Would you choose to deposit your money for one year at 3% per annum
or for five years at 3% per annum? The chances are that you would
choose one year because this gives you more financial flexibility. It ties up
your funds for a shorter period of time.
Table 1.6 Example of rates offered by a bank to
its customers.
Maturity
(years)
Deposit
rate
Mortgage
rate
1
5
3%
3%
6%
6%
21
Introduction
Now suppose that you want a mortgage. Again you agree with the
prevailing view that interest rate increases are just as likely as interest rate
decreases. Would you choose a one-year mortgage at 6% or a five-year
mortgage at 6%? The chances are that you would choose a five-year
mortgage because it fixes your borrowing rate for the next five years and
subjects you to less refinancing risk.
When the bank posts the rates shown in Table 1.6, it is likely to find
that the majority of its depositors opt for one-year maturities and the
majority of its customers seeking mortgages opt for five-year maturities.
This creates an asset/liability mismatch for the bank and subjects its net
interest income to risks. There is no problem if interest rates fall. The
bank will find itself financing the five-year 6% mortgages with deposits
that cost less than 3% and net interest income will increase. However, if
rates rise, the deposits that are financing these 6% mortgages will cost
more than 3% and net interest income will decline. A 3% rise in interest
rates would reduce the net interest income to zero.
It is the job of the asset/liability management group to ensure that the
maturities of the assets on which interest is earned and the maturities of the
liabilities on which interest is paid are matched. One way to do this in our
example is to increase the five-year rate on both deposits and mortgages.
For example, we could move to the situation in Table 1.7 where the fiveyear deposit rate is 4% and the five-year mortgage rate 7%. This would
make five-year deposits relatively more attractive and one-year mortgages
relatively more attractive. Some customers who chose one-year deposits
when the rates were as in Table 1.6 will switch to five-year deposits when
rates are as in Table 1.7. Some customers who chose five-year mortgages
when the rates were as in Table 1.6 will choose one-year mortgages. This
may lead to the maturities of assets and liabilities being matched. If there is
still an imbalance with depositors tending to choose a one-year maturity
and borrowers a five-year maturity, five-year deposit and mortgage rates
could be increased even further. Eventually the imbalance will disappear.
The net result of all banks behaving in the way we have just described is
Table 1.7 Five-year rates are increased in an attempt
to match maturities of assets and liabilities.
Maturity
(years)
Deposit
rate
Mortgage
rate
1
5
3%
4%
6%
7%
22
Chapter 1
Business Snapshot 1.2 Expensive Failures of Financial Institutions in the US
Throughout the 1960s, 1970s, and 1980s, Savings and Loans (S&Ls) in the
United States failed to manage interest rate risk well. They tended to take
short-term deposits and make long-term fixed-rate mortgages. As a result they
were seriously hurt by interest rate increases in 1966, 1969 70, 1974. and the
killer in 1979-82. S&Ls were protected by government guarantees. Over 1,700
failed in the 1980s. A major reason for the failures was their inadequate
interest rate risk management. The total cost to the US taxpayer of the failures
has been estimated to be between $100 and $500 billion.
The largest bank failure in the US, Contintental Illinois, can also be
attributed to a failure to manage interest rale risks well. During the period
1980 to 1983 its assets (i.e., loans) with maturities over a year totaled between
$7 billion and $8 billion, whereas its liabilities (i.e., deposits) with maturities
over a year were between $1.4 billion and $2.5 billion. Continental failed in
1984 and was the subject of an expensive government bailout.
that long-term rates tend to be higher than those predicted by expected
future short-term rates. This phenomenon is referred to as liquidity
preference theory. It leads to long-term rates being higher than short-term
rates most of the time. Even when the market expects a small decline in
short-term rates, liquidity preference theory is likely to cause long-term
rates to be higher than short-term rates.
Many banks now have sophisticated systems for monitoring the decisions being made by customers so that, when they detect small differences
between the maturities of the assets and liabilities being chosen, they can
fine-tune the rates they offer. Sometimes derivatives such as interest rate
swaps are also used to manage their exposure. The result is that net interest
income is very stable and does not lead to significant risks. However, as
indicated in Business Snapshot 1.2, this has not always been the case.
SUMMARY
An important general principle in finance is that there is a trade-off
between risk and return. Higher expected returns can usually be achieved
only by taking higher risks. Investors should not, in theory, be concerned
with risks they can diversify away. The extra return they demand should
be for the amount of nondiversifiable systematic risk they are bearing.
For companies, investment decisions are more complicated. Companies are not in general as well diversified as investors and survival is an
Introduction
23
important and legitimate objective. Both financing and investment
decisions should be taken so that the possibility of financial distress
is low.
This is because financial distress leads to what are known as bankruptcy costs. These costs arise from the nature of the bankruptcy process
and almost invariably lead to a reduction in shareholder value over and
above the reduction that took place as a result of the adverse events
leading to bankruptcy.
Banks must manage the risks they face carefully. Equity capital is
typically about 5% of assets and profit before taxes is often less than
1% of assets. Large trading losses, an economic downturn leading to a
sharp rise in loan losses, or other unexpected events can lead to an
erosion of equity capital and put the bank in a precarious position.
Regulators have become increasingly active in ensuring that the capital
a bank keeps is commensurate with the risks it takes.
Two general approaches to risk management are risk decomposition
and risk aggregation. Risk decomposition involves managing risks one
by one. Risk aggregation involves relying on the power of diversification
to reduce risks. Banks use both approaches to manage market risks.
Credit risks have traditionally been managed using risk aggregation, but
with the advent of credit derivatives the risk decomposition approach
can be used.
A bank's net interest income is the excess of the interest earned over the
interest paid. There are now well established asset/liability management
procedures to ensure that this remains roughly constant from year to year.
These involve adjusting the rates offered to customers to ensure that the
maturities of assets and liabilities are matched.
FURTHER READING
Markowitz, H. "Portfolio Selection," Journal of Finance, 1, 1 (March 1952),
77-91.
Ross, S. "The Arbitrage Theory of Capital Asset Pricing," Journal of Economic
Theory, 13 (December 1976), 343-362.
Sharpe, W. "Capital Asset Prices: A Theory of Market Equilibrium," Journal of
Finance, September 1964, 425-442.
Smith, C. W., and R. M. Stulz, "The Determinants of a Firm's Hedging Policy,"
Journal of Financial and Quantitative Analysis, 20 (1985), 391-406.
Stulz, R. M., Risk Management and Derivatives. Southwestern, 2003.
24
Chapter 1
QUESTIONS A N D PROBLEMS (Answers at End of Book)
1.1. An investment has probabilities 0.1, 0.2, 0.35, 0.25, and 0.1 of giving
returns equal to 40%, 30%, 15%, - 5 % and - 1 5 % . What is the expected
return and the standard deviation of returns?
1.2. Suppose that there are two investments with the same probability distribution of returns as in Problem 1.1. The correlation between the
returns is 0.15. What is the expected return and standard deviation of
return from a portfolio where money is divided equally between the
investments.
1.3. For the two investments considered in Figure 1.2 and Table 1.2, what are
the alternative risk/return combinations if the correlation is (a) 0.3, (b) 1.0,
and (c) -1.0.
1.4. What is the difference between systematic and nonsystematic risk? Which
is more important to an equity investor? Which can lead to the bankruptcy
of a corporation?
1.5. Outline the arguments leading to the conclusion that all investors should
choose the same portfolio of risky investments. What are the key
assumptions?
1.6. The expected return on the market portfolio is 12% and the risk-free rate
is 6%. What is the expected return on an investment with a beta of (a) 0.2,
(b) 0.5, and (c) 1.4?
1.7. "Arbitrage pricing theory is an extension of the capital asset pricing
model." Explain this statement.
1.8. "The capital structure decision of a company is a trade-off between
bankruptcy costs and the tax advantages of debt." Explain this statement.
1.9. A bank's operational risk is the risk of large losses because of employee
fraud, natural disasters, litigation, etc. It will be discussed in Chapter 14. Is
operational risk best handled by risk decomposition or risk aggregation.
1.10. A bank's profit next year will be normally distributed with a mean of 0.6%
of assets and a standard deviation of 1.5% of assets. The bank's equity is
4% of assets. What is the probability that the bank will have a positive
equity at the end of the year? Ignore taxes.
1.11. Why do you think that banks are regulated to ensure that they do not take
too much risk but most other companies (e.g., those in manufacturing and
retailing) are not?
1.12. Explain carefully the risks faced by Continental Illinois in the 1980 to 1983
period based on the data in Business Snapshot 1.2.
1.13. Explain carefully why interest rate risks contributed to the expensive S&L
failures in the United States.
Introduction
25
1.14. Suppose that a bank has $5 billion of one-year loans and $20 billion of
five-year loans. These are financed by $15 billion of one-year deposits and
$10 billion of five-year deposits. Explain the impact on the bank's net
interest income of interest rates increasing by 1 % every year for the next
three years.
1.15. List the bankruptcy costs incurred by the company in Business Snapshot 1.1.
ASSIGNMENT QUESTIONS
1.16. Suppose that one investment has a mean return of 8% and a standard
deviation of return of 14%. Another investment has a mean return of 12%
and a standard deviation of return of 20%. The correlation between the
returns is 0.3. Produce a chart similar to Figure 1.2 showing alternative
risk/return combinations from the two investments.
1.17. Which items on a DLC's income statement in Table 1.4 are most likely to
be affected by (a) credit risk, (b) market risk, and (c) operational risk.
1.18. A bank estimates that its profit next year is normally distributed with a
mean of 0.8% of assets and the standard deviation of 2% of assets. How
much equity (as a percent of assets) does the company need to be (a) 99%
and (b) 99.9% sure that it will have positive equity at the end of the year.
Ignore taxes.
1.19. Suppose that a bank has $10 billion of one-year loans and $30 billion of
five-year loans. These are financed by $35 billion of one-year deposits and
$5 billion of five-year deposits. The bank has equity totaling $2 billion and
its return on equity is currently 12%. Estimate what change in the interest
rates next year would lead to the bank's return on equity being reduced to
zero. Assume that the bank is subject to a tax rate of 30%.
1.20. Explain why long-term rates are higher than short-term rates most of the
time. Under what circumstances would you expect long-term rates to be
lower than short-term rates?
Financial Products
and How They Are
Used for Hedging
Companies trade a variety of financial instruments to manage their risks.
Some of these instruments are referred to as standard or "plain vanilla"
products. Most forward contracts, futures contracts, swaps, and options
fall into this category. Others are designed to meet the particular needs of
a corporate treasurer. These are referred to as "exotics" or structured
products. This chapter describes the instruments and how they trade. It
discusses the circumstances when a company should hedge, how much
hedging it should do, and what instruments should be used.
2.1 THE MARKETS
There are two types of markets in which financial instruments trade.
These are known as the exchange-traded market and the over-the-counter
(or OTC) market.
Exchange-Traded Markets
Exchanges have been used to trade financial products for many years.
Some exchanges such as the New York Stock Exchange (NYSE) focus on
the trading of stocks. Others such as the Chicago Board of Trade (CBOT)
and the Chicago Board Options Exchange (CBOE) are concerned with
the trading of derivatives such as futures and options.
28
Chapter 2
The role of the exchange is to define the contracts that trade and
organize trading so that market participants can be sure that the trades
they agree to will be honored. Traditionally individuals have met at the
exchange and agreed on the prices for trades, often by using an elaborate
system of hand signals. Exchanges are increasingly moving to electronic
trading. This involves traders entering their desired trades at a keyboard
and a computer being used to match buyers and sellers. Not everyone
agrees that the shift to electronic trading is desirable. Electronic trading is
less physically demanding than traditional floor trading. However, traders
do not have the opportunity to attempt to predict short-term market
trends from the behavior and body language of other traders.
Sometimes trading is facilitated with market makers. These are individuals who are always prepared to quote both a bid price (the price at
which they are prepared to buy) and an offer price (the price at which they
are prepared to sell). Typically the exchange will specify an upper bound
for the spread between a market maker's bid and offer prices.
Over-the-Counter Markets
The over-the-counter market is an important alternative to exchanges. It
is a telephone- and computer-linked network of traders who work for
financial institutions, large corporations, or fund managers. Financial
institutions often act as market makers for the more commonly traded
instruments.
Telephone conversations in the over-the-counter market are usually
taped. If there is a dispute over what was agreed, the tapes are replayed
to resolve the issue. Trades in the over-the-counter market are typically
much larger than trades in the exchange-traded market. A key advantage
of the over-the-counter market is that the terms of a contract do not have
to be those specified by an exchange. Market participants are free to
negotiate any mutually attractive deal. A disadvantage is that there is
usually some credit risk in an over-the-counter trade (i.e., there is a small
risk that the contract will not be honored). Exchanges have organized
themselves to eliminate virtually all credit risk.
2.2 WHEN TO HEDGE
Most nonfinancial companies have no particular skills or expertise in
predicting variables such as interest rates, exchange rates, and commodity prices. It makes sense for them to hedge the risks associated with
Financial Products and How They Are Used for Hedging
29
these variables as they arise. The companies can then focus on their main
activities. By hedging, they avoid unpleasant surprises such as a foreign
exchange loss or a sharp rise in the price of a commodity that has to be
purchased.
It can be argued that companies need not hedge because the company's
shareholders can implement their own hedging programs, deciding which
of the company's risks to keep and which to get rid of. However this
assumes—unrealistically—that a company's shareholders have as much
information about the risks faced by the company as the company's
management. It also ignores the bankruptcy costs arguments in
Section 1.2.
Hedging and Competitors
It is not always correct for a company to choose to hedge. If hedging is
not the norm in a certain industry, it can be dangerous for one company
to choose to be different from all others. Competitive pressures within the
industry may be such that the prices of the goods and services produced
by the industry fluctuate to reflect raw material costs, interest rates,
exchange rates, and so on. A company that does not hedge can expect
its profit margins to be roughly constant. However, a company that does
hedge can expect its profit margins to fluctuate!
To illustrate this point, consider two manufacturers of gold jewelry,
SafeandSure Company and TakeaChance Company. We assume that
most companies in the industry do not hedge against movements in the
price of gold and that TakeaChance Company is no exception. However,
SafeandSure Company has decided to be different from its competitors
and to use futures contracts to lock in the price it will pay for gold over
the next 18 months.
If the price of gold goes up, economic pressures will tend to lead to a
corresponding increase in the wholesale price of the jewelry, so that
TakeaChance Company's profit margin is unaffected. By contrast,
SafeandSure Company's profit margin will increase after the effects of
the hedge have been taken into account. If the price of gold goes down,
economic pressures will tend to lead to a corresponding decrease in the
wholesale price of the jewelry. Again, TakeaChance Company's profit
margin is unaffected. However, SafeandSure Company's profit margin
goes down. In extreme conditions, SafeandSure Company's profit margin
could become negative as a result of the "hedging" carried out! This
example is summarized in Table 2.1.
Chapter 2
30
Table 2.1
Danger in hedging when competitors do not.
Change in
gold price
Effect on price
of gold jewelry
Effect on profits
of TakeaChance Co.
Effect on profits
of SafeandSure Co.
Increase
Decrease
Increase
Decrease
None
None
Increase
Decrease
The example emphasizes the importance of looking at the big picture
when hedging. All the implications of changes in commodity prices,
interest rates, and exchange rates on a company's profitability should
be taken into account in the design of a hedging strategy.
2.3 THE "PLAIN VANILLA" PRODUCTS
In this section we review the products that most commonly trade in
financial markets. We focus on those that involve stocks, currencies,
commodities, and interest rates. Less traditional products such as credit
derivatives, weather derivatives, energy derivatives, and insurance derivatives are covered in later chapters.
*
Long and Short Positions in Assets
The simplest type of trade is the purchase or sale of an asset. Examples of
such trades are:
1.
2.
3.
4.
The
The
The
The
purchase of 100 IBM shares
sale of 1 million British pounds
purchase of 1000 ounces of gold
sale of $1 million worth of bonds issued by General Motors
The first of these trades would typically be done on an exchange; the
other three would be done in the over-the-counter market. The trades are
sometimes referred to as spot contracts because they lead to almost
immediate "on the spot" delivery of the asset.
Short Sales
In some markets it is possible to sell an asset that you do not own with
the intention of buying it back later. This is referred to as shorting the
asset. We will illustrate how it works by considering a short sale of shares
of a stock.
Financial Products and How They Are Used for Hedging
31
Suppose an investor instructs a broker to short 500 IBM shares. The
broker will carry out the instructions by borrowing the shares from
another client and selling them on an exchange in the usual way. The
investor can maintain the short position for as long as desired, provided
there are always shares available for the broker to borrow. At some stage,
however, the investor will close out the position by purchasing 500 IBM
shares. These are then replaced in the account of the client from which
the shares were borrowed. The investor takes a profit if the stock price
has declined and a loss if it has risen. If, at any time while the contract is
open, the broker runs out of shares to borrow, the investor is shortsqueezed and is forced to close out the position immediately, even if not
ready to do so.
An investor with a short position must pay to the broker any income,
such as dividends or interest, that would normally be received on the
securities that have been shorted. The broker will transfer this to the client
account from which the securities have been borrowed. Consider the
position of an investor who shorts 500 shares in April when the price
per share is $120 and closes out the position by buying them back in July
when the price per share is $100. Suppose that a dividend of $1 per share
is paid in May. The investor receives 500 x $120 = $60,000 in April when
the short position is initiated. The dividend leads to a payment by the
investor of 500 x $1 = $500 in May. The investor also pays
500 x $100 = $50,000 for shares when the position is closed out in July.
The net gain is, therefore,
$60,000 - $500 - $50,000 = $9,500
Table 2.2 illustrates this example and shows that the cash flows from the
short sale are the mirror image of the cash flows from purchasing the
shares in April and selling them in July.
An investor entering into a short position is required to maintain a
margin account with the broker. The margin account consists of cash or
marketable securities deposited by the investor with the broker to guarantee that the investor will not walk away from the short position if the
share price increases. An initial margin is deposited and, if there are
adverse movements (i.e., increases) in the price of the asset that is being
shorted, additional margin may be required. The margin account does
not represent a cost to the investor. This is because interest is usually paid
on the balance in margin accounts and, if the interest rate offered is
unacceptable, marketable securities such as Treasury bills can be used to
Chapter 2
32
Table 2.2
Cash flows from short sale and purchase of shares.
Purchase of Shares
April: Purchase 500 shares
shares for
for $120
$120
May: Receive dividend
dividend
July: Sell 500 shares for
for $100
$100 per
per share
share
Net profit = -$9,
- $ 9 , 500
-$60,000
+$500
+$50,000
Short Sale of
Short
ofShares
Shares
April: Borrow 500 shares and sell
sell them
them for
for $120
$120
+$60,000
May: Pay dividend
dividend
-$500
-$500
July: Buy 500 shares for
for $100
$100 per
per share
share
-$50,000
Replace borrowed shares to close short position
Net profit = +$9, 500
meet margin requirements. The proceeds of the sale of the asset belong to
the investor and normally form part of the initial margin.
Forward Contracts
A forward contract is an agreement to buy an asset in the future for a
certain price. Forward contracts trade in the over-the-counter market.
One of the parties to a forward contract assumes a long position and
agrees to buy the underlying asset on a certain specified future date for a
certain specified price. The other party assumes a short position and
agrees to sell the asset on the same date for the same price.
Forward contracts on foreign exchange are very popular. Table 2.3
provides quotes on the exchange rate between the British pound (GBP)
and the US dollar (USD) that might be provided by a large international
Table 2.3 Spot and forward quotes for the
USD/GBP exchange rate, August 5, 2005 (GBP =
British pound; USD = US dollar; quote is number
of USD per GBP).
Spot
1-month forward
3-month forward
6-month forward
Bid
Offer
1.7794
1.7780
1.7761
1.7749
1.7798
1.7785
1.7766
1.7755
Financial Products and How They Are Used for Hedging
33
bank on August 5, 2005. The quotes are for the number of USD per GBP.
The first row indicates that the bank is prepared to buy GBP (also known
as sterling) in the spot market (i.e., for virtually immediate delivery) at the
rate of $1.7794 per GBP and sell sterling in the spot market at $1.7798 per
GBP; the second row indicates that the bank is prepared to buy sterling in
one month's time at $1.7780 per GBP and sell sterling in one month at
$1.7785 per GBP; and so on.
Forward contracts can be used to hedge foreign currency risk. Suppose
that on August 5, 2005, the treasurer of a US corporation knows that the
corporation will pay £1 million in six months (on February 5, 2006) and
wants to hedge against exchange rate moves. The treasurer can agree to
buy £1 million six months forward at an exchange rate of 1.7755 by
trading with the bank providing the quotes in Table 2.3. The corporation
then has a long forward contract on GBP. It has agreed that on February
5, 2006, it will buy £1 million from the bank for $1.7755 million. The
bank has a short forward contract on GBP. It has agreed that on
February 5, 2006, it will sell £1 million for $1.7755 million. Both the
corporation and the bank have made a binding commitment.
What are the possible outcomes in the trade we have just described?
The forward contract obligates the corporation to buy £1 million for
$1,775,500 and the bank to sell £1 million for this amount. If the spot
exchange rate rose to, say, 1.8000 at the end of the six months the forward
contract would be worth +$24,500 (= $1,800,000 - $1,775,500) to the
corporation and -$24,500 to the bank. It would enable 1 million pounds
to be purchased at 1.7755 rather than 1.8000. Similarly, if the spot
exchange rate fell to 1.6000 at the end of the six months, the forward
contract would have a value of -$175,500 to the corporation and a value
of +$175,500 to the bank because it would lead to the corporation paying
$175,500 more than the market price for the sterling.
In general, the payoff from a long position in a forward contract on one
unit of an asset is
ST — K
where K is the delivery price and ST is the spot price of the asset at
maturity of the contract. This is because the holder of the contract is
obligated to buy an asset worth ST for K. Similarly, the payoff from a
short position in a forward contract on one unit of an asset is
K — ST
These payoffs can be positive or negative. They are illustrated in Figure 2.1.
34
Chapter 2
Figure 2.1 Payoffs from forward contracts: (a) long position and (b) short
position. Delivery price = K; price of asset at contract maturity = ST.
Because it costs nothing to enter into a forward contract, the payoff from
the contract is also the trader's total gain or loss from the contract.
Futures Contracts
Futures contracts like forward contracts are agreements to buy an asset at
a future time. Unlike forward contracts, futures are traded on an exchange. This means that the contracts that trade are standardized. The
exchange defines the amount of the asset underlying one contract, when
delivery can be made, exactly what can be delivered, and so on. Contracts
are referred to by their delivery month. For example the September 2007
gold futures is a contract where delivery is made in September 2007. (The
precise times, delivery locations, etc., are defined by the exchange.)
One of the attractive features of futures contracts is that it is easy to close
out a position. If you buy (i.e., take a long position in) a September gold
futures contract in March you can exit in June by selling (i.e., taking a short
position in) the same contract. In forward contracts final delivery of the
underlying asset is usually made. Futures contracts by contrast are usually
closed out before the delivery month is reached. Business Snapshot 2.1 is
an amusing story indicating a potential pitfall in closing out contracts.
Futures contracts are different from forward contracts in that they are
settled daily. If the futures price moves in your favor during a day, you
make an immediate gain. If it moves in the opposite direction, you make
Financial Products and How They Are Used for Hedging
Business Snapshot 2.1
35
The Unanticipated Delivery of a Futures Contract
This story (which may well be apocryphal) was told to the author of this book
by a senior executive of a financial institution. It concerns a new employee of the
financial institution who had not previously worked in the financial sector. One
of the clients of the financial institution regularly entered into a long futures
contract on live cattle for hedging purposes and issued instructions to close out
the position on the last day of trading. (Live cattle futures contracts trade on the
Chicago Mercantile Exchange and each contract is on 40.000 pounds of cattle.)
The new employee was given responsibility for handling the account.
When the time came to close out a contract, the employee noted that the
client was long one contract and instructed a trader at the exchange to go long
(not short) one contract. The result of this mistake was that the financial
institution ended up with a long position in two live cattle futures contracts.
By the time the mistake was spotted, trading in the contract had ceased.
The financial institution (not the client) was responsible for the mistake. As
a result, it started to look into the details of the delivery arrangements for live
cattle futures contracts something it had never done before. Under the terms
of the contract, cattle could be delivered by the party with the short position
to a number of different locations in the United States during the delivery
month. Because it was long the financial institution could do nothing but wait
for a party with a short position to issue a notice of intention to deliver to the
exchange and for the exchange to assign that notice to the financial institution.
It eventually received a notice from the exchange and found that it would
receive live cattle at a location 2,000 miles away the following Tuesday. The new
employee was dispatched to the location to handle things. It turned out that the
location had a cattle auction every Tuesday. The party with the short position
that was making delivery bought cattle at the auction and then immediately
delivered them. Unfortunately, the cattle could not be resold until the next cattle
auction the following Tuesday. The employee was therefore faced with the
problem of making arrangements for the cattle to be housed and fed for a
week. This was a great start to a first job in the financial sector!
an immediate loss. Consider what happens when you buy one September gold futures contract on the Chicago Board of Trade when the
futures price is $580 per ounce. The contract is on 100 ounces of gold.
You must maintain a margin account with your broker. As in the case
of a short sale, this consists of cash or marketable securities and is to
ensure that you will honor your commitments under the contract. The
rules for determining the initial amount that must be deposited in a
margin account, when it must be topped up, and so on, are set by the
exchange.
36
Chapter 2
Suppose that the initial margin requirement on your gold trade is
$2000. If, by close of trading on the first day you hold the contract,
the September gold futures price has dropped from $580 to $578, then
you lose 2 x 100 or $200. This is because you agreed to buy gold in
September for $580 and the going price for September gold is now $578.
The balance in your margin account would be reduced from $2,000 to
$1,800. If, at close of trading the next day, the September futures price is
$577, then you lose a further $100 from your margin account. If the
decline continues, you will at some stage be asked to add cash to your
margin account. If you do not do so, your broker will close out your
position.
In the example we have just considered, the price of gold moved against
you. If instead it moved in your favor, funds would be added to your
margin account. The Exchange Clearinghouse is responsible for managing the flow of funds from investors with short positions to investors
with long positions when the futures price increases and the flow of funds
in the opposite direction when the futures price declines.
The relationship between futures or forward prices and spot prices is
given in Appendix A at the end of the book.
Swaps
The first swap contracts were negotiated in the early 1980s. Since then the
market has seen phenomenal growth. Swaps now occupy a position of
central importance in the over-the-counter derivatives market.
A swap is an agreement between two companies to exchange cash flows
in the future. The agreement defines the dates when the cash flows are to
be paid and the way in which they are to be calculated. Usually the
calculation of the cash flows involves the future values of interest rates,
exchange rates, or other market variables.
A forward contract can be viewed as a simple example of a swap.
Suppose it is March 1, 2007, and a company enters into a forward
contract to buy 100 ounces of gold for $600 per ounce in one year. The
company can sell the gold in one year as soon as it is received. The
forward contract is therefore equivalent to a swap where the company
agrees that on March 1, 2008, it will pay $60,000 and receive 1005,
where S is the market price of one ounce of gold on that date.
Whereas a forward contract is equivalent to the exchange of cash flows
on just one future date, swaps typically lead to cash flow exchanges
taking place on several future dates. The most common swap is a "plain
Financial Products and How They Are Used for Hedging
37
5%
Company A
Counterparty
LIBOR
Figure 2.2 A plain vanilla interest rate swap.
vanilla" interest rate swap where a fixed rate of interest is exchanged for
LIBOR.1 Both interest rates are applied to the same notional principal.
A swap where company A pays a fixed rate of interest of 5% and receives
LIBOR is shown in Figure 2.2. Suppose that in this contract interest
rates are reset every six months, the notional principal is $100 million,
and the swap lasts for three years. Table 2.4 shows the cash flows to
company A when six-month LIBOR interest rates prove to be those
shown in the second column of the table. The swap is entered into on
March 5, 2007. The six-month interest rate on that date is 4.2% per year
or 2.1 % per six months. As a result, the floating-rate cash flow received
six months later on September 5, 2007, is 0.021 x 100 or $2.1 million.
Similarly, the six month interest rate of 4.8% per annum (or 2.4% per six
months) on September 5, 2007, leads to the floating cash flow received
six months later (on March 5, 2008) being $2.4 million, and so on. The
fixed-rate cash flow paid is always $2.5 million (5% of $100 million
Table 2.4 Cash flows (millions of dollars) to company A in swap
in Figure 2.2. The swap lasts three years and has a principal of
Date
Floating
Net
6-month $100 million.
Fixed
Mar.
Sept.
Mar.
Sept.
Mar.
Sept.
Mar.
5,
5,
5,
5,
5,
5,
5,
2007
2007
2008
2008
2009
2009
2010
LIBOR rate
(%)
cash f l o w
received
cash flow
paid
cash flow
4.20
4.80
5.30
5.50
5.60
5.90
6.40
+2.10
+2.40
+2.65
+2.75
+2.80
+2.95
-2.50
-2.50
-2.50
-2.50
-2.50
-2.50
-0.40
-0.10
+0.15
+0.25
+0.30
+0.45
1
LIBOR is the London Interbank Offered Rate. It is the rate at which a bank offers to
make large wholesale deposits with another bank and will be discussed in Chapter 4.
Chapter 2
38
applied to a six-month period). 2 Note that the timing of cash flows
corresponds to the usual way short-term interest rates such as LIBOR
work. The interest is observed at the beginning of the period to which it
applies and paid at the end of the period.
Plain vanilla interest rate swaps are very popular because they can be
used for many purposes. For example, the swap in Figure 2.2 could be
used by company A to transform borrowings at a floating rate of LIBOR
plus 1% to borrowings at a fixed rate of 6%. (Pay LIBOR plus 1%,
receive LIBOR, and pay 5% nets out to pay 6%.) It can also be used by
company A to transform an investment earning a fixed rate of 4.5% to an
investment earning LIBOR minus 0.5%. (Receive 4.5%, pay 5%, and
receive LIBOR nets out to receive LIBOR minus 0.5%.)
Example 2.1
Suppose a bank has floating-rate deposits and five-year fixed-rate loans. As
explained in Section 1.5, this exposes the bank to significant risks. If rates rise,
then the deposits will be rolled over at high rates and the bank's net interest
income will contract. The bank can hedge its risks by entering into the swap in
Figure 2.2 (taking the role of Company A). The swap can be viewed as
transforming the floating-rate deposits to fixed-rate deposits. (Alternatively,
it can be viewed as transforming fixed-rate loans to floating-rate loans.)
Many banks are market makers in swaps. Table 2.5 shows quotes for US
dollar swaps that might be posted by a bank. 3 The first row shows that the
Table 2.5
2
Swap quotes made by a market maker
(percent per annum).
Maturity
(years)
Bid
Offer
Swap rate
2
3
4
5
7
10
6.03
6.21
6.35
6.47
6.65
6.83
6.06
6.24
6.39
6.51
6.68
6.87
6.045
6.225
6.370
6.490
6.665
6.850
Note that we have not taken account of day count conventions, holidays, calendars,
etc., in Table 2.4.
3
The standard swap in the United States is one where fixed payments made every six
months are exchanged for floating LIBOR payments made every three months. In
Table 2.4 we assumed that fixed and floating payments are exchanged every six months.
Financial Products and How They Are Used for Hedging
Business Snapshot 2.2
39
Procter and Gamble's Bizarre Deal
A particularly bizarre swap is the so-called "5/30" swap entered into between
Bankers Trust (BT) and Procter and Gamble (P&G) on November 2, 1993.
This was a five-year swap with semiannual payments. The notional principal
was $200 million. BT paid P&G 5.30% per annum. P&G paid BT the average
30-day CP (commercial paper) rate minus 75 basis points plus a spread. The
average commercial paper rate was calculated by taking observations on the
30-day commercial paper rate each day during the preceding accrual period
and averaging them.
The spread was zero for the first payment date (May 2, 1994). For the
remaining nine payment dates, it was
In this, five-year CMT is the constant maturity Treasury yield (i.e., the yield on
a five-year Treasury note, as reported by the US Federal Reserve). The 30-year
TSY price is the midpoint of the bid and offer cash bond prices for the 6.25%
Treasury bond maturing on August 2023. Note that the spread calculated from
the formula is a decimal interest rate. It is not measured in basis points. If the
formula gives 0.1 and the CP rate is 6%, the rate paid by P&G is 15.25%.
P&G was hoping that the spread would be zero and the deal would enable it
to exchange fixed-rate funding at 5.30% for funding at 75 basis points less than
the commercial paper rate. In fact, interest rates rose sharply in early 1994, bond
prices fell, and the swap proved very, very expensive. (See Problem 2.30.)
bank is prepared to enter into a two-year swap where it pays a fixed rate of
6.03% and receives LIBOR. It is also prepared to enter into a swap where it
receives 6.06% and pays LIBOR. The bid-offer spread in Table 2.5 is three
or four basis points. The average of the bid and offer fixed rates is known as
the swap rate. This is shown in the final column of the table.
The trading of swaps is facilitated by ISDA, the International Swaps
and Derivatives Association. This organization has developed standard
contracts that are widely used by market participants. Swaps can be
designed so that the periodic cash flows depend on the future value of
any well-defined variable. Swaps dependent on interest rates, exchange
rates, commodity prices, and equity indices are popular. Sometimes there
are embedded options in a swap. For example, one side might have the
option to terminate a swap early or to choose between a number of
40
Chapter 2
different ways of calculating cash flows. Occasionally swaps are traded
with payoffs that are calculated in quite bizarre ways. An example is a
deal entered into between Procter and Gamble and Bankers Trust in 1993
(see Business Snapshot 2.2). The details of this transaction are in the
public domain because it later became the subject of litigation.4 The
valuation of swaps is discussed in Appendix B at the end of the book.
Options
Options are traded both on exchanges and in the over-the-counter
market. There are two basic types of options. A call option gives the
holder the right to buy the underlying asset by a certain date for a certain
price. A put option gives the holder the right to sell the underlying asset
by a certain date for a certain price. The price in the contract is known as
the exercise price or strike price; the date in the contract is known as the
expiration date or maturity. American options can be exercised at any time
up to the expiration date, but European options can be exercised only on
the expiration date itself.5 Most of the options that are traded on
exchanges are American. In the exchange-traded equity option market,
one contract is usually an agreement to buy or sell 100 shares. European
options are generally easier to analyze than American options, and some
of the properties of an American option are frequently deduced from
those of its European counterpart.
An at-the-money option is an option where the strike price is close to
the price of the underlying asset. An out-of-the-money option is a call
option where the strike price is above the price of the underlying asset or
a put option where the strike price is below this price. An in-the-money
option is a call option where the strike price is below the price of the
underlying asset or a put option where the strike price is above this price.
It should be emphasized that an option gives the holder the right to do
something. The holder does not have to exercise this right. By contrast, in
a forward or futures contract, the holder is obligated to buy or sell the
underlying asset. Note that, whereas it costs nothing to enter into a
forward or futures contract, there is a cost to acquiring an option.
The largest exchange in the world for trading stock options is the
Chicago Board Options Exchange (CBOE; www.cboe.com). Table 2.6
4
See D.J. Smith, "Aggressive Corporate Finance: A Close Look at the Procter and
Gamble-Bankers Trust Leveraged Swap." Journal of Derivatives 4, No. 4 (Summer 1997):
67-79.
5
Note that the terms American and European do not refer to the location of the option
or the exchange. Some options trading on North American exchanges are European.
Financial Products and How They Are Used for Hedging
Table 2.6
41
Prices of options on Intel, May 29, 2003; stock price = $20.83.
Calls
Strike price
($)
June
July
Oct.
June
July
Oct.
20.00
22.50
1.25
0.20
1.60
0.45
2.40
1.15
0.45
1.85
0.85
2.20
1.50
2.85
Puts
gives the closing prices of some of the American options trading on Intel
on May 29, 2003. The option strike prices are $20 and $22.50. The
maturities are June 2003, July 2003, and October 2003. The June options
have an expiration date of June 21, 2003; the July options have an
expiration date of July 19, 2003; the October options have an expiration
date of October 18, 2003. Intel's stock price at the close of trading on
May 29, 2003, was $20.83.
Suppose an investor instructs a broker to buy one October call option
contract on Intel with a strike price of $22.50. The broker will relay these
instructions to a trader at the CBOE. This trader will then find another
trader who wants to sell one October call contract on Intel with a strike
price of $22.50, and a price will be agreed upon. We assume that the price
is $1.15, as indicated in Table 2.6. This is the price for an option to buy
one share. In the United States, one stock option contract is a contract to
buy or sell 100 shares. Therefore the investor must arrange for $115 to be
remitted to the exchange through the broker. The exchange will then
arrange for this amount to be passed on to the party on the other side of
the transaction.
In our example the investor has obtained at a cost of $115 the right to
buy 100 Intel shares for $22.50 each. The party on the other side of the
transaction has received $115 and has agreed to sell 100 Intel shares for
$22.50 per share if the investor chooses to exercise the option. If the price of
Intel does not rise above $22.50 before October 18, 2003, the option is not
exercised and the investor loses $115. But if the Intel share price does well
and the option is exercised when it is $30, the investor can buy 100 shares at
$22.50 per share when they are worth $30 per share. This leads to a gain of
$750, or $635 when the initial cost of the options is taken into account.
An alternative trade for the investor would be the purchase of one July
put option contract with a strike price of $20. From Table 2.6 we see that
this would cost 100 x 0.85 or $85. The investor would obtain at a cost of
$85 the right to sell 100 Intel shares for $20 per share prior to July 19,2003.
If the Intel share price stays above $20 the option is not exercised and the
Chapter 2
42
Profit ($)
Figure 2.3 Net profit per share from (a) purchasing a contract consisting of 100
Intel October call options with a strike price of $22.50 and (b) purchasing a
contract consisting of 100 Intel July put options with a strike price of $20.00.
investor loses $85. But if the investor exercises when the stock price is $15,
the investor makes a gain of $500 by buying 100 Intel shares at $15 and
selling them for $20. The net profit after the cost of the options is taken into
account is $415.
The options trading on the CBOE are American. If we assume for
simplicity that they are European so that they can be exercised only at
maturity, the investor's profit as a function of the final stock price for the
Intel options we have been considering is shown in Figure 2.3.
There are four types of trades in options markets:
1.
2.
3.
4.
Buying a call
Selling a call
Buying a put
Selling a put
Buyers are referred to as having long positions; sellers are referred to as
having short positions. Selling an option is also known as writing the
option.
Options trade very actively in the over-the-counter market as well as on
exchanges. Indeed the over-the-counter market for options is now larger
than the exchange-traded market. Whereas exchange-traded options tend
to be American, options trading in the over-the-counter market are
frequently European. The advantage of the over-the-counter market is
that maturity dates, strike prices, and contract sizes can be tailored to
meet the precise needs of a client. They do not have to correspond to
those specified by the exchange. Option trades in the over-the-counter
market are usually much larger than those on exchanges.
Financial
Products and How They Are Used for Hedging
43
Valuation formulas and numerical procedures for options on stocks,
stock indices, currencies, and futures are in Appendices C and D at the
end of this book.
2.4 USING THE PRODUCTS FOR HEDGING
Futures and forward contracts provide a hedge for an exposure at one
particular time. As we saw earlier the treasurer of a US company can use
the quotes in Table 2.3 to buy sterling forward when it is known that the
company will have to pay sterling at a certain future time. Similarly, the
treasurer can use the quotes to sell sterling forward when it is known that
it will receive sterling at a certain future time. Futures contracts can be
used in a similar way. When a futures contract is used for hedging the
plan is usually to close the contract out prior to maturity. As a result the
hedge performance is reduced somewhat because there is uncertainty
about the difference between the futures price and the spot price on the
close-out date. This uncertainty is known as basis risk.
When forward and futures contracts are used for hedging the objective
is to lock in the price at which an asset will be bought or sold at a certain
future time. The hedge ratio is the ratio of the size of the futures or
forward position to the size of the exposure. Up to now we have assumed
that a company uses a hedge ratio of 1.0. (For example, if it has a
$1 million exposure to the USD/GBP exchange rate it takes a $1 million
forward or futures position.) Sometimes a company may choose to
partially hedge its risks by using a hedge ratio of less than 1.0.
Even when a company wants to minimize its risks, it may not be
optimal for it to use a hedge ratio of 1.0. Suppose that the standard
deviation of the change in a futures or forward price during the hedging
period is
and the standard deviation of the change in the value of the
asset being hedged is
Suppose further that the correlation between the
two changes is
It can be shown that the optimal hedge ratio is
(2.1)
Example 2.2
An airline expects to purchase 2.4 million gallons of jet fuel in one month's
time. Because there is no futures contract on jet fuel it decides to use the
futures contact on heating oil that trades on the New York Mercantile
Exchange. The correlation between monthly changes in the price of jet fuel
and monthly changes in heating oil futures price is 0.7. The standard deviation
Chapter 2
44
of monthly changes in the heating oil futures price per gallon is 0.024 and the
standard deviation of the monthly changes in the price of jet oil per gallon is
0.021. The optimal hedge ratio is therefore
Each heating oil futures contract is on 42,000 gallons of heating oil. The
number of contracts the company should buy is therefore
Example 2.3
A fund manager wants to hedge a well-diversified investment portfolio worth
$2.5 million until time T using a forward contracts on the S&P 500. The index
is currently 1250, so that the portfolio is worth 2000 times the index. Assume
that F is the current forward rate for a contract maturing at time T. If the
portfolio has a beta of 1.0, the hedge ratio should be 1.0 (see Section 1.1 for a
discussion of beta). This means that the forward contract should be structured
so that the payoff to the fund manager at time T is
2000(F - ST)
where ST is the value of the S&P 500 at time T. When
position should be doubled, so that the payoff is
= 2, the hedge
4000(F - ST)
In general, the hedge ratio should equal the beta
of the well-diversified
portfolio, so that the payoff from the forward contract is
This
is consistent with equation (2.1) because it is approximately true that
and = 1.
As explained earlier, a swap can be regarded as a convenient way of
bundling forward contracts. It can provide a hedge for cash flows that will
occur on a regular basis over a period of time.
Options are a different type of hedging instrument from forwards,
futures, and swaps. Whereas forward, futures, and swap contracts lock
in prices for future sales or purchases of an asset, an option provides
insurance. For example, the call option in Figure 2.3a could be used to
guarantee that shares of Intel could be purchased for $22.50 or less in
October; the put option in Figure 2.3b could be used to guarantee that a
holding of shares in Intel could be sold for at least $20 in July.
A Practical Issue
It is important to realize that hedging can result in a decrease or an
increase in a company's profits relative to the position it would be in with
Financial
Products
and
How
They
Are
Used
for
Hedging
45
no hedging. Consider a company that decides to use a short futures
position to hedge the future sale of 1 million barrels of oil. If the price
of oil goes down, the company loses money on the sale of the oil and the
futures position leads to an offsetting gain. The treasurer can be congratulated for having had the foresight to put the hedge in place. Clearly,
the company is better off than it would be with no hedging. Other
executives in the organization, it is hoped, will appreciate the contribution
made by the treasurer.
If the price of oil goes up, the company gains from its sale of the oil,
and the futures position leads to an offsetting loss. The company is in a
worse position than it would be with no hedging. Although the hedging
decision was perfectly logical, the treasurer may in practice have a difficult
time justifying it. Suppose that the price of oil increases by $3, so that the
company loses about $3 per barrel on the futures contract. We can
imagine a conversation such as the following between the treasurer and
the president:
PRESIDENT:
TREASURER:
PRESIDENT:
TREASURER :
PRESIDENT :
TREASURER:
PRESIDENT :
TREASURER :
This is terrible. We've lost $3 million in the futures
market in the space of three months. How could it
happen? I want a full explanation.
The purpose of the futures contracts was to hedge our
exposure to the price of oil—not to make a profit. Don't
forget that we made about $3 million from the favorable
effect of the oil price increases on our business.
What's that got to do with it? That's like saying that we
do not need to worry when our sales are down in
California because they are up in New York.
If the price of oil had gone down...
I don't care what would have happened if the price of oil
had gone down. The fact is that it went up. I really do not
know what you were doing playing the futures markets
like this. Our shareholders will expect us to have done
particularly well this quarter. I'm going to have to
explain to them that your actions reduced profits by
$3 million. I'm afraid this is going to mean no bonus
for you this year.
That's unfair. I was only...
Unfair! You are lucky not to be fired. You lost $3 million.
It all depends how you look at i t . . .
46
Chapter 2
This shows that, although hedging reduces risk for the company, it can
increase risk for the treasurer if others do not fully understand what is
being done. The only real solution to this problem is to ensure that all
senior executives within the organization fully understand the nature of
hedging before a hedging program is put in place. One of the reasons why
treasurers sometimes choose to buy insurance using options rather than
implementing a more straightforward hedge using forwards, futures, or
swaps is that options do not lead to the problem we have just mentioned.
They allow the company to benefit from favorable outcomes while being
hedged against unfavorable outcomes. (Of course, this is achieved at a
cost. The company has to pay the option premium.)
2.5 EXOTIC OPTIONS AND STRUCTURED DEALS
We met one exotic swap transaction in Business Snapshot 2.2. Many
different types of exotic options and highly structured deals trade in the
over-the-counter market. Although exotics are a relatively small part of
the trading for a financial institution they are important because the profit
margin on trades in exotics tends to be much higher than on plain vanilla
options or swaps. Here are a few examples of exotic options:
Asian Options: Whereas regular options provide a payoff based on the
final price of the underlying asset at the time of exercise, Asian options
provide a payoff based on the average of the price of the underlying asset
over some specified period. An example is an average price call option that
provides a payoff in one year equal to max(S - K, 0), where S is the
average asset price during the year and K is the strike price.
Barrier Options: These options come into existence or disappear when the
price of the underlying asset reaches a certain barrier. For example, a
knock-out call option with a strike price of $30 and a barrier of $20 is a
regular call option that ceases to exist if the asset price falls below $20.
Basket Options: These are options on a portfolio of assets rather than
options on a single asset.
Binary Options: These are options that provide a fixed dollar payoff if
some criterion is met. An example is an option that provides a payoff in
one year of $1,000 if a stock price is greater than $20.
Compound Options: These are options on options. There are four types: a
call on a call, a call on a put, a put on a call, and a put on a put. An example
of a compound option is an option to buy an option on a stock currently
Financial Products and How They Are Used for Hedging
Business Snapshot 2.3
47
Microsoft's Hedging
Microsoft actively manages its foreign exchange exposure. In some countries
(e.g., Europe. Japan, and Australia) it bills in the local currency and converts
jits net revenue to US dollars monthly. For these currencies there is a clear
exposure to exchange rate movements. In other countries (e.g., Latin America,
Eastern Europe. and Southeast Asia) it bills in US dollars. The latter appears
to avoid any foreign exchange exposure - but it does not.
Suppose the US dollar strengthens against the currency of a country where
Microsoft is billing in dollars. People in the country will find it more difficult
to buy Microsoft products because it takes more of the local currency to buy
11. As a result Microsoft will probably find it necessary to reduce its US dollar
prices or face a decline in sales. Microsoft therefore has a foreign exchange
exposure—both when it bills in US dollars and when it bills in the local
currency. This emphasizes the point made in Section 2.2 that it is important to
consider the big picture when hedging.
Microsoft likes to use options for hedging. Suppose it uses a one-year time
horizon. Microsoft recognizes that its exposure to. say, the Japanese yen is an
exposure to the average exchange rate during the year because approximately
the same amount of yen is converted to US dollars each month. It therefore uses
Asian options rather than regular options to hedge the exposure. What is more,
Microsoft's net exposure is to a weighted average of the exchange rates for all
the countries in which it does business. It therefore uses basket options (i.e.,
options on a weighted average of exchange rates). A contract it likes to negotiate
with financial institutions is therefore an Asian basket put option. This cost of
this option is much less than a portfolio of put options, one for each month and
each exchange rate (see Problem 2.24). But it gives Microsoft exactly the
protection it requires.
Microsoft faces other financial risks. For example, it is exposed to interest
rate risk on its bond portfolio. (When rates rise, the portfolio loses money.) It
also has two sorts of exposure to equity prices. It is exposed to the equity
prices of the companies in which it invests. It is also exposed to its own equity
price because it regularly repurchases its own shares as part of its stock awards
program. It likes to use sophisticated option strategies to hedge these risks.
worth $15. The first option expires in one year and has a strike price of $1.
The second option expires in three years and has a strike price of $20.
Lookback Options: These are options that provide a payoff based on the
maximum or minimum price of the underlying asset over some period.
An example is an option that provides a payoff in one year equal to
ST - Smin, where ST is the asset price at the end of the year and Smin is the
minimum asset price during the year.
48
Chapter 2
Why do companies use exotic options and structured products in
preference to the plain vanilla products we looked at in Section 2.3?
Sometimes the products are totally inappropriate as risk management
tools. (This was certainly true in the case of the Procter and Gamble
swap discussed in Business Snapshot 2.2.) But usually there are sound
reasons for the contracts entered into by corporate treasurers. For
example, Microsoft often uses Asian basket options in its risk management. As explained in Business Snapshot 2.3 this is the ideal product for
managing its exposures.
2.6 DANGERS
Derivatives are very versatile instruments. They can be used for hedging,
for speculation, and for arbitrage. (Hedging involves reducing risks;
speculation involves taking risks; arbitrage involves locking in a profit
by simultaneously trading in two or more markets.) It is this very
versatility that can cause problems. Sometimes traders who have a
mandate to hedge risks or follow an arbitrage strategy become (consciously or unconsciously) speculators. The results can be disastrous.
One example of this is provided by the activities of Nick Leeson at
Barings Bank (see Business Snapshot 2.4).6
To avoid the problems Barings encountered, it is very important for
both financial and nonfinancial corporations to set up controls to ensure
that derivatives are being used for their intended purpose. Risk limits
should be set and the activities of traders monitored daily to ensure that
the risk limits are adhered to.
SUMMARY
There are two types of markets in which financial products trade: the
exchange-traded market and the over-the-counter market. In this chapter
we have reviewed spot trades, forward contracts, futures contracts, swaps,
and options contracts. A forward or futures contract involves an obligation to buy or sell an asset at a certain time in the future for a certain
price. A swap is an agreement to exchange cash flows in the future in
amounts dependent on the values of one or more market variables. There
are two types of options: calls and puts. A call option gives the holder the
6
The movie Rogue Trader provides a good dramatization of the failure of Barings Bank.
Financial Products and How They Are Used for Hedging
Business Snapshot 2.4
49
The Barings Bank Disaster
Derivatives are very versatile instruments. They can be used for hedging,
speculation, and arbitrage. One of the risks faced by a company that trades
derivatives is that an employee who has a mandate to hedge or to look for
arbitrage opportunities may become a speculator.
Nick Leeson, an employee of Barings Bank in the Singapore office in 1995.
had a mandate to look for arbitrage opportunities between the Nikkei 225
futures prices on the Singapore exchange and the Osaka exchange. Over time
Leeson moved from being an arbitrageur to being a speculator without anyone
in Barings head office in London fully understanding that he had changed the
way he was using derivatives. He began to make losses, which he was able to
hide. He then began to take bigger speculative positions in an attempt to
recover the losses, but only succeeded in making the losses worse.
In the end Leeson's total loss was close to 1 billion dollars. As a result.
Barings a bank that had been in existence for 200 years -was wiped out.
One of the lessons from the Barings disaster is that it is important to define
unambiguous risk limits for traders and then carefully monitor their activities
to make sure that the limits are adhered to.
right to buy an asset by a certain date for a certain price. A put option
gives the holder the right to sell an asset by a certain date for a certain
price. Forwards, futures, swaps, and options trade on a wide range of
different underlying assets.
Forward, futures, and swap contracts have the effect of locking in the
prices that will apply to future transactions. Options by contrast provide
insurance. They ensure that the price applicable to a future transaction
will not be worse than a certain level. Exotic options and structured
products are tailored to the particular needs of corporate treasurers.
For example, as we saw in Business Snapshot 2.3, Asian basket options
can allow a company such as Microsoft to hedge its net exposure to
several risks over a period of time.
It is important to look at the big picture when hedging. For example,
a company may find that it is increasing rather than reducing its risks if
it chooses to hedge when none of its competitors does so. The hedge
ratio is the ratio of the size of the hedge position to the size of the
exposure. It is not always optimal to use a hedge ratio of 1.0. The
optimal hedge ratio depends on the variability of futures price, the
variability of the price of the asset being hedged, and the correlation
between the two.
50
Chapter 2
FURTHER READING
Baz, J., and M. Pascutti. "Alternative Swap Contracts Analysis and Pricing,"
Journal of Derivatives, Winter 1996: 7-21.
Boyle, P., and F. Boyle, Derivatives: The Tools That Changed Finance. London:
Risk Books, 2001
Brown, K. C. and D. J. Smith. Interest Rate and Currency Swaps: A Tutorial.
Association for Investment Management and Research, 1996.
Brown, G. W. "Managing Foreign Exchange Risk with Derivatives." Journal of
Financial Economics, 60 (2001): 401-448.
Flavell, R. Swaps and Other Instruments. Chichester: Wiley, 2002.
Geczy, C, B.A. Minton, and C. Schrand. "Why Firms Use Currency
Derivatives," Journal of Finance, 52, No. 4 (1997): 1323-1354.
Litzenberger, R. H. "Swaps: Plain and Fanciful," Journal of Finance, 47, No. 3
(1992): 831-850.
Miller, M. H. "Financial Innovation: Achievements and Prospects," Journal of
Applied Corporate Finance, 4 (Winter 1992): 4-11.
Warwick B., F. J. Jones, and R. J. Teweles. The Futures Game, 3rd edn. New
York: McGraw-Hill, 1998.
QUESTIONS A N D PROBLEMS (Answers at End of Book)
2.1. What is the difference between a long forward position and a short
forward position?
2.2. Explain the difference between hedging, speculation, and arbitrage.
2.3. What is the difference between entering into a long forward contract when
the forward price is $50 and taking a long position in a call option with a
strike price of $50?
2.4. Explain carefully the difference between selling a call option and buying a
put option.
2.5. An investor enters into a short forward contract to sell 100,000 British
pounds for US dollars at an exchange rate of 1.5000 US dollars per
pound. How much does the investor gain or lose if the exchange rate at
the end of the contract is (a) 1.4900 and (b) 1.5200?
2.6. A trader enters into a short cotton futures contract when the futures price
is 50 cents per pound. The contract is for the delivery of 50,000 pounds.
How much does the trader gain or lose if the cotton price at the end of the
contract is (a) 48.20 cents per pound and (b) 51.30 cents per pound?
Financial
Products and How They Are Used for Hedging
51
2.7. Suppose you write a put contract with a strike price of $40 and an
expiration date in three months. The current stock price is $41 and the
contract is on 100 shares. What have you committed yourself to? How
much could you gain or lose?
2.8. What is the difference between the over-the-counter market and the
exchange-traded market? Which of the two markets do the following trade
in: (a) a forward contract, (b) a futures contract, (c) an option, (d) a swap,
and (e) an exotic option?
2.9. You would like to speculate on a rise in the price of a certain stock. The
current stock price is $29, and a three-month call with a strike of $30 costs
$2.90. You have $5,800 to invest. Identify two alternative strategies, one
involving an investment in the stock and the other involving investment in
the option. What are the potential gains and losses from each?
2.10. Suppose that you own 5,000 shares worth $25 each. How can put options
be used to provide you with insurance against a decline in the value of
your holding over the next four months?
2.11. When first issued, a stock provides funds for a company. Is the same true
of a stock option? Discuss.
2.12. Suppose that a March call option to buy a share for $50 costs $2.50 and is
held until March. Under what circumstances will the holder of the option
make a profit? Under what circumstances will the option be exercised?
2.13. Suppose that a June put option to sell a share for $60 costs $4 and is held
until June. Under what circumstances will the seller of the option (i.e., the
party with the short position) make a profit? Under what circumstances
will the option be exercised?
2.14. A company knows that it is due to receive a certain amount of a foreign
currency in four months. What type of option contract is appropriate for
hedging?
2.15. A United States company expects to have to pay 1 million Canadian
dollars in six months. Explain how the exchange rate risk can be hedged
using (a) a forward contract and (b) an option.
2.16. In the 1980s, Bankers Trust developed index currency option notes
(ICONs). These are bonds in which the amount received by the holder
at maturity varies with a foreign exchange rate. One example was its trade
with the Long Term Credit Bank of Japan. The ICON specified that if the
yen/US dollar exchange rate, ST, is greater than 169 yen per dollar at
maturity (in 1995), the holder of the bond receives $1,000. If it is less than
169 yen per dollar, the amount received by the holder of the bond is
52
Chapter 2
When the exchange rate is below 84.5, nothing is received by the holder at
maturity. Show that this ICON is a combination of a regular bond and two
options.
2.17. Suppose that USD/GBP spot and forward exchange rates are as follows:
Spot
90-day forward
180-day forward
1.6080
1.6056
1.6018
What opportunities are open to an arbitrageur in the following situations:
(a) a 180-day European call option to buy £1 for $1.57 costs 2 cents and
(b) a 90-day European put option to sell £1 for $1.64 costs 2 cents?
2.18. A company has money invested at 5% for five years. It wishes to use the
swap quotes in Table 2.5 to convert its investment to a floating-rate
investment. Explain how it can do this.
2.19. A company has borrowed money for five years at 7%. Explain how it can
use the quotes in Table 2.5 to convert this to a floating-rate liability.
2.20. A company has a has a floating-rate liability that costs LIBOR plus 1%.
Explain how it can use the quotes in Table 2.5 to convert this to a threeyear fixed-rate liability.
2.21. A corn farmer argues: "I do not use futures contracts for hedging. My real
risk is not the price of corn. It is that my whole crop gets wiped out by the
weather." Discuss this viewpoint. Should the farmer estimate his or her
expected production of corn and hedge to try to lock in a price for
expected production?
2.22. An airline executive has argued: "There is no point in our hedging the
price of jet fuel. There is just as much chance that we will lose from doing
this as that we will gain." Discuss the executive's viewpoint.
2.23. The standard deviation of monthly changes in the spot price of live cattle
is (in cents per pound) 1.2. The standard deviation of monthly changes in
the futures price of live cattle for the closest contract is 1.4. The correlation
between the futures price changes and the spot price changes is 0.7. It is
now October 15. A beef producer is committed to purchasing 200,000
pounds of live cattle on November 15. The producer wants to use the
December live-cattle futures contracts to hedge its risk. Each contract is
for the delivery of 40,000 pounds of cattle. What strategy should the beef
producer follow?
2.24. Why is the cost of an Asian basket put option to Microsoft considerably
less than the cost of a portfolio of put options, one for each currency and
each maturity (see Business Snapshot 2.3.)?
Financial Products and How They Are Used for Hedging
53
ASSIGNMENT QUESTIONS
2.25. The current price of a stock is $94, and three-month European call options
with a strike price of $95 currently sell for $4.70. An investor who feels
that the price of the stock will increase is trying to decide between buying
100 shares and buying 2,000 call options (= 20 contracts). Both strategies
involve an investment of $9,400. What advice would you give? How high
does the stock price have to rise for the option strategy to be more
profitable?
2.26. A bond issued by Standard Oil worked as follows. The holder received no
interest. At the bond's maturity the company promised to pay $1,000 plus
an additional amount based on the price of oil at that time. The additional
amount was equal to the product of 170 and the excess (if any) of the price
of a barrel of oil at maturity over $25. The maximum additional amount
paid was $2,550 (which corresponds to a price of $40 per barrel). Show
that the bond is a combination of a regular bond, a long position in call
options on oil with a strike price of $25, and a short position in call
options on oil with a strike price of $40.
2.27. The price of gold is currently $500 per ounce. The forward price for
delivery in one year is $700. An arbitrageur can borrow money at 10%
per annum. What should the arbitrageur do? Assume that the cost of
storing gold is zero and that gold provides no income.
2.28. A company's investments earn LIBOR minus 0.5%. Explain how it can
use the quotes in Table 2.5 to convert the investments to (a) 3-year,
(b) 5-year, and (c) 10-year fixed-rate investments.
2.29. What position is equivalent to a long forward contract to buy an asset at
K on a. certain date and a long position in a European put option to sell it
for K on that date.
2.30. Estimate the interest rate paid by P&G on the 5/30 swap in Business
Snapshot 2.2 if (a) the CP rate is 6.5% and the Treasury yield curve is flat
at 6% and (b) the CP rate is 7.5% and the Treasury yield curve is flat at
7% with semiannual compounding.
2.31. It is July 16. A company has a portfolio of stocks worth $100 million. The
beta of the portfolio is 1.2. The company would like to use the CME
December futures contract on the S&P 500 to change the beta of the
portfolio to 0.5 during the period July 16 to November 16. The index is
currently 1,000, and each contract is on $250 times the index, (a) What
position should the company take? (b) Suppose that the company changes
its mind and decides to increase the beta of the portfolio from 1.2 to 1.5.
What position in futures contracts should it take?
How Traders
Manage Their
Exposures
The trading function within a financial institution is referred to as the front
office; the part of the financial institution that is concerned with the overall
level of the risks being taken, capital adequacy, and regulatory compliance
is referred to as the middle office; the record keeping function is referred to
as the back office. As explained in Chapter 1, there are two levels within a
financial institution at which trading risks are managed. First, the front
office hedges risks by ensuring that exposures to individual market variables are not too great. Second, the middle office aggregates the exposures
of all traders to determine whether the total risk is acceptable. In this
chapter we focus on the hedging activities of the front office. In later
chapters we will consider how risks are aggregated in the middle office.
This chapter explains what are termed the "Greek letters", or simply
the "Greeks". Each of the Greeks measures a different aspect of the risk
in a trading position. Traders calculate their Greeks at the end of each day
and are required to take action if the internal risk limits of the financial
institution they work for are exceeded. Failure to take this action is liable
to lead to immediate dismissal.
3.1 DELTA
Imagine that you are a trader working for a US bank and responsible for
all trades involving gold. The current price of gold is $500 per ounce.
56
Chapter 3
Table 3.1
Summary of gold portfolio.
Position
Value ($)
Spot gold
Forward Contracts
Futures Contracts
Swaps
Options
Exotics
180,000
-60,000
2,000
80,000
-110,000
25,000
Total
117,000
Table 3.1 shows a summary of your portfolio (known as your "book").
How can you manage your risks?
The value of your portfolio is currently $117,000. One way of investigating the risks you face is to revalue the portfolio on the assumption that
there is a small increase in the price of gold from $500 per ounce to
$500.10 per ounce. Suppose that the new value of the portfolio is
$116,900. A $0.10 increase in the price of gold decreases the value of
your portfolio by $100. The sensitivity of the portfolio to the price of
gold is therefore
This is referred to as the delta of the portfolio. The portfolio loses value
at a rate of $1,000 per $1 increase in the price of gold. Similarly, it gains
value at a rate of $1,000 per $1 decrease in the price of gold.
In general, the delta of a portfolio with respect to a market variable is
where
is a small change in the value of the variable and
is the
resultant change in the value of the portfolio. Using calculus terminology,
delta is the partial derivative of the portfolio value with respect to the
value of the variable:
In our example the trader can eliminate the delta exposure by buying
1,000 ounces of gold. This is because the delta of a position in 1,000 ounces
of gold is 1,000. (The position gains value at the rate of $1,000 per $1
How Traders
Manage
Their
Exposures
57
increase in the price of gold.) When this trade is combined with the existing
portfolio, the resultant portfolio has a delta of zero. Such a portfolio is
referred to as a delta-neutral portfolio.
Linear Products
A linear product is a product whose value at any given time is linearly
dependent on the value of the underlying asset price (see Figure 3.1).
Forward contracts, futures contracts, and swaps are linear products;
options are not.
A linear product can be hedged relatively easily. Consider, for example,
a US bank that enters into a forward contract with a corporate client
where it agrees to sell the client 1 million euros at a certain exchange rate
in one year. Assume that the euro interest rate is 4% with annual
compounding. This means that the present value of 1 million euros in
one year is 961,538 euros. The bank can hedge its risk by borrowing
enough dollars to buy 961,538 euros today and then investing the euros
for one year at 4%. The bank knows that it will have the 1 million euros it
needs to deliver in one year and it knows what its costs will be.
When the bank enters into the opposite transaction and agrees to buy
1 million euros in one year it must hedge by shorting 961,538 euros. It
Figure 3.1
A linear product
Chapter 3
58
Business Snapshot 3.1
Hedging by Gold Mining Companies
It is natural for a gold mining company to consider hedging against changes
in the price of gold. Typically it takes several years to extract all the gold from
a mine. Once a gold mining company decides to go ahead with production at
a particular mine, it has a big exposure to the price of gold. Indeed, a mine
that looks profitable at the outset could become unprofitable if the price of
gold plunges.
Gold mining companies are careful to explain their hedging strategies to
potential shareholders. Some gold mining companies do not hedge. They tend
to attract shareholders who buy gold stocks because they want to benefit
when the price of gold increases and are prepared to accept the risk of a loss
from a decrease in the price of gold. Other companies choose to hedge. They
estimate the number of ounces they will produce each month for the next few
years and enter into short futures or forward contracts to lock in the price
that will be received.
Suppose you are Goldman Sachs and have just entered into a forward
contract with a gold mining company where you agree to buy a large amount
of gold at a fixed price. How do you hedge your risk? The answer is that you
borrow gold from a central bank and sell it at the current market price. (The
central banks of many countries hold large amounts of gold.) At the end of
the life of the forward contract, you buy gold from the gold mining company
under the terms of the forward contract and use it to repay the central bank.
The central bank charges a fee (perhaps 1.5% per annum) known as the gold
lease rate for lending its gold in this way.
does this by borrowing the euros today at 4% and immediately converting
them to US dollars. The 1 million euros received in one year are used to
repay the loan.
Shorting assets to hedge forward contracts is not always easy. Gold is
an interesting case in point. Financial institutions often find that they
enter into very large forward contracts to buy gold from gold producers.
This means that they need to borrow large quantities of gold to create a
short position for hedging. As outlined in Business Snapshot 3.1, central
banks are the source of the borrowed gold.
Nonlinear Products
Options and most structured products are nonlinear products. The
relationship between the value of the product and the value of the
underlying market variable at any given time is nonlinear. This nonlinearity makes them more difficult to hedge.
How Traders Manage Their Exposures
59
Asset price
Figure 3.2
Value of option as a function of stock price.
Consider as an example a trader who sells 100,000 European call
options on a non-dividend-Paying stock when
1.
2.
3.
4.
5.
The
The
The
The
The
stock price is $49.
strike price is $50.
risk-free interest rate is 5%.
stock price volatility is 20% per annum.
time to option maturity is 20 weeks.
We suppose that the amount received for the options is $300,000 and that
the trader has no other positions dependent on the stock.
The value of one option as a function of the underlying stock price is
shown in Figure 3.2. The delta of one option changes with the stock price
in the way shown in Figure 3.3.1 At the time of the trade, the value of an
option to buy one share of the stock is $2.40 and the delta of the option is
0.522. Because the trader is short 100,000 options, the value of the
trader's portfolio is -$240,000 and the delta of the portfolio is
—52,200. The trader can feel pleased that the options have been sold
for $60,000 more than their theoretical value, but is faced with the
problem of hedging the risk in the position.
The portfolio can be made delta neutral immediately after the trade by
1
Figures 3.2 and 3.3 were produced with the DerivaGem software, which can be
downloaded from the author's website. The Black-Scholes (analytic) model is used.
60
Chapter 3
Asset price
Figure 3.3
Delta of option as a function of stock price.
buying 52,200 shares of the underlying stock. If there is a small decrease
(increase) in the stock price, the gain (loss) on the option position should
be offset by the loss (gain) on the shares. For example, if the stock price
increases from $49 to $49.10, then the value of the options will decrease by
52,200 x 0.1 = $5,220 while that of the shares will increase by this amount.
In the case of linear products, once the hedge has been set up, it does
not need to be changed. This is not the case for nonlinear products. To
preserve delta neutrality, the hedge has to be adjusted periodically. This is
known as rebalancing.
Tables 3.2 and 3.3 provide two examples of how rebalancing might
work in our example. Rebalancing is assumed to be done weekly. As
mentioned, the initial value of delta for a single option is 0.522 and the
delta of the portfolio is —52,200. This means that, as soon as the option is
written, $2,557,800 must be borrowed to buy 52,200 shares at a price of
$49. The rate of interest is 5%. An interest cost of approximately $2,500 is
therefore incurred in the first week.
In Table 3.2 the stock price falls by the end of the first week to $48.12.
The delta declines to 0.458. A long position in 45,800 shares is now
required to hedge the option position. A total of 6,400 (i.e., 52,200 45,800) shares is therefore sold to maintain the delta neutrality of the
hedge. The strategy realizes $308,000 in cash, and the cumulative borrowings at the end of Week 1 are reduced to $2,252,300. During the second
week, the stock price reduces to $47.37 and delta declines again. This
How Traders Manage Their Exposures
Table 3.2
61
Simulation of delta hedging. Option closes in the money and cost of
hedging is $263,300.
Week
Stock
price
Delta
Shares
purchased
Cost of shares
purchased
($000)
Cumulative cash
outflow
($000)
Interest
cost
($000)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
49.00
48.12
47.37
50.25
51.75
53.12
53.00
51.87
51.38
53.00
49.88
48.50
49.88
50.37
52.13
51.88
52.87
54.87
54.62
55.87
57.25
0.522
0.458
0.400
0.596
0.693
0.774
0.771
0.706
0.674
0.787
0.550
0.413
0.542
0.591
0.768
0.759
0.865
0.978
0.990
1.000
1.000
52,200
(6,400)
(5,800)
19,600
9,700
8,100
(300)
(6,500)
(3,200)
11,300
(23,700)
(13,700)
12,900
4,900
17,700
(900)
10,600
11,300
1,200
1,000
0
2,557.8
(308.0)
(274.7)
984.9
502.0
430.3
(15.9)
(337.2)
(164.4)
598.9
(1,182.2)
(664.4)
643.5
246.8
922.7
(46.7)
560.4
620.0
65.5
55.9
0.0
2,557.8
2,252.3
1,979.8
2,966.6
3,471.5
3,905.1
3,893.0
3,559.5
3,398.5
4,000.7
2,822.3
2,160.6
2,806.2
3,055.7
3,981.3
3,938.4
4,502.6
5,126.9
5,197.3
5,258.2
5,263.3
2.5
2.2
1.9
2.9
3.3
3.8
3.7
3.4
3.3
3.8
2.7
2.1
2.7
2.9
3.8
3.8
4.3
4.9
5.0
5.1
leads to 5,800 shares being sold at the end of the second week. During the
third week, the stock price increases to over $50 and delta increases. This
leads to 19,600 shares being purchased at the end of the third week.
Toward the end of the life of the option, it becomes apparent that the
option will be exercised and delta approaches 1.0. By Week 20, therefore,
the hedger owns 100,000 shares. The hedger receives $5 million (i.e.,
100,000 x $50) for these shares when the option is exercised so that
the total cost of writing the option and hedging it is $263,300.
Table 3.3 illustrates an alternative sequence of events where the option
closes out of the money. As it becomes clear that the option will not be
exercised, delta approaches zero. By Week 20 the hedger therefore has no
Position in the underlying stock. The total costs incurred are $256,600.
Chapter 3
62
Table 3.3
Simulation of delta hedging. Option closes out of the money ;
cost of hedging is $256,600.
Week
Stock
price
Delta
Shares
purchased
Cost of shares
purchased
($000)
Cumulative cash
outflow
($000)
Interest
cost
($000)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
49.00
49.75
52.00
50.00
48.38
48.25
48.75
49.63
48.25
48.25
51.12
51.50
49.88
49.88
48.75
47.50
48.00
46.25
48.13
46.63
48.12
0.522
0.568
0.705
0.579
0.459
0.443
0.475
0.540
0.420
0.410
0.658
0.692
0.542
0.538
0.400
0.236
0.261
0.062
0.183
0.007
0.000
52,200
4,600
13,700
(12,600)
(12,000)
(1,600)
3,200
6,500
(12,000)
(1,000)
24,800
3,400
(15,000)
(400)
(13,800)
(16,400)
2,500
(19,900)
12,100
(17,600)
(700)
2,557.8
228.9
712.4
(630.0)
(580.6)
(77.2)
156.0
322.6
(579.0)
(48.2)
1,267.8
175.1
(748.2)
(20.0)
(672.7)
(779.0)
120.0
(920.4)
582.4
(820.7)
(33.7)
2,557.8
2,789.2
3,504.3
2,877.7
2,299.9
2,224.9
2,383.0
2,707.9
2,131.5
2,085.4
3,355.2
3,533.5
2,788.7
2,771.4
2,101.4
1,324.4
1,445.7
526.7
1,109.6
290.0
256.6
2.5
2.7
3.4
2.8
2.2
2.1
2.3
2.6
2.1
2.0
3.2
3.4
2.7
2.7
2.0
1.3
1.4
0.5
1.1
0.3
In Tables 3.2 and 3.3 the costs of hedging the option, when discounted
to the beginning of the period, are close to but not exactly the same as
the theoretical (Black-Scholes) price of $240,000. If the hedging scheme
worked perfectly, the cost of hedging would, after discounting, be exactly
equal to the Black-Scholes price for every simulated stock price path.
The reason for the variation in the cost of delta hedging is that the hedge
is rebalanced only once a week. As rebalancing takes place more frequently, the variation in the cost of hedging is reduced. Of course, the
examples in Tables 3.2 and 3.3 are idealized in that they assume the
model underlying the Black-Scholes formula is exactly correct and there
are no transaction costs.
Delta hedging aims to keep the value of the financial institution's
How Traders Manage Their Exposures
63
position as close to unchanged as possible. Initially, the value of the
written option is $240,000. In the situation depicted in Table 3.2, the value
of the option can be calculated as $414,500 in Week 9. Thus, the financial
institution has lost $174,500 (i.e., 414,500 - 240,000) on its short option
position. Its cash position, as measured by the cumulative cost, is
$1,442,900 worse in Week 9 than in Week 0. The value of the shares held
has increased from $2,557,800 to $4,171,100 for a gain of $1,613,300. The
net effect of all this is that the value of the financial institution's position
has changed by only $4,100 during the nine-week period.
Where the Cost Comes From
The delta-hedging scheme in Tables 3.2 and 3.3 in effect creates a long
position in the option synthetically to neutralize the trader's short option
position. As the tables illustrate, the scheme tends to involve selling stock
just after the price has gone down and buying stock just after the price
has gone up. It might be termed a buy-high, sell-low scheme! The cost of
$240,000 comes from the average difference between the price paid for the
stock and the price realized for it.
Transaction Costs
Maintaining a delta-neutral position in a single option and the underlying
asset, in the way that has just been described, is liable to be prohibitively
expensive because of the transaction costs incurred on trades. Delta
neutrality is more feasible for a large portfolio of derivatives dependent
on a single asset. Only one trade in the underlying asset is necessary to
zero out delta for the whole portfolio. The hedging transactions costs are
absorbed by the profits on many different trades.
3.2 GAMMA
The gamma, , of a portfolio of options on an underlying asset is the rate
of change of the portfolio's delta with respect to the price of the underlying asset. It is the second partial derivative of the portfolio with respect
to asset price:
If gamma is small, then delta changes slowly and adjustments to keep a
Portfolio delta neutral only need to be made relatively infrequently.
Chapter 3
64
Figure 3.4
Hedging error introduced by nonlinearity.
However, if gamma is large in absolute terms, then delta is highly sensitive
to the price of the underlying asset. It is then quite risky to leave a deltaneutral portfolio unchanged for any length of time. Figure 3.4 illustrates
this point. When the stock price moves from S to S', delta hedging
assumes that the option price moves from C to C', when in fact it moves
from C to C". The difference between C' and C" leads to a hedging error.
This error depends on the curvature of the relationship between the option
price and the stock price. Gamma measures this curvature.2
Gamma is positive for a long position in an option. The general way in
which gamma varies with the price of the underlying asset is shown in
Figure 3.5. Gamma is greatest for options where the stock price is close to
the strike price, K.
Making a Portfolio Gamma Neutral
A linear product has zero gamma and cannot be used to change the gamma
of a portfolio. What is required is a position in an instrument, such as an
option, that is not linearly dependent on the underlying asset price.
Suppose that a delta-neutral portfolio has a gamma equal to and a
traded option has a gamma equal to
If the number of traded options
added to the portfolio is
the gamma of the portfolio is
Hence, the position in the traded option necessary to make the portfolio
2
Indeed, the gamma of an option is sometimes referred to as its curvature by
practitioners.
How Traders Manage Their Exposures
65
Figure 3.5 Relationship between gamma of an option and
price of underlying asset. K is the option's strike price.
gamma neutral is
Including the traded option is likely to
change the delta of the portfolio, so the position in the underlying asset
then has to be changed to maintain delta neutrality. Note that the
portfolio is gamma neutral only for a short period of time. As time
passes, gamma neutrality can be maintained only if the position in the
traded option is adjusted so that it is always equal to
Making a delta-neutral portfolio gamma neutral can be regarded as a
first correction for the fact that the position in the underlying asset cannot
be changed continuously when delta hedging is used. Delta neutrality
provides protection against relatively small stock price moves between
rebalancing. Gamma neutrality provides protection against larger movements in this stock price between hedge rebalancing. Suppose that a
portfolio is delta neutral and has a gamma of -3,000. The delta and
gamma of a particular traded call option are 0.62 and 1.50, respectively.
The portfolio can be made gamma neutral by including in the portfolio a
long position of 3,000/1.5 = 2,000 in the call option. However, the delta
of the portfolio will then change from zero to 2,000 x 0.62 = 1,240. A
quantity, 1,240, of the underlying asset must therefore be sold to keep it
delta neutral.
3.3 VEGA
Another source of risk in derivatives trading is volatility. The volatility of
a market variable measures our uncertainty about the future value of the
variable. (It will be discussed more fully in Chapter 5.) In option
Chapter 3
66
valuation models, volatilities are often assumed to be constant, but in
practice they do change through time. Spot positions, forwards, and
swaps do not depend on the volatility of the underlying market variable,
but options and most exotics do. Their values are liable to change
because of movements in volatility as well as because of changes in the
asset price and the passage of time.
The vega,
of a portfolio is the rate of change of the value of the
portfolio with respect to the volatility of the underlying market variable.3
If vega is high in absolute terms, the portfolio's value is very sensitive to
small changes in volatility. If vega is low in absolute terms, volatility
changes have relatively little impact on the value of the portfolio.
The vega of a portfolio can be changed by adding a position in a traded
option. If V is the vega of the portfolio and
is the vega of a traded
option, a position of
in the traded option makes the portfolio
instantaneously vega neutral. Unfortunately, a portfolio that is gamma
neutral will not, in general, be vega neutral, and vice versa. If a hedger
requires a portfolio to be both gamma and vega neutral, then at least two
traded derivatives dependent on the underlying asset must usually be
used.
Example 3.1
Consider a portfolio that is delta neutral, with a gamma of —5,000 and a vega
of —8,000. A traded option has a gamma of 0.5, a vega of 2.0, and a delta of
0.6. The portfolio could be made vega neutral by including a long position in
4,000 traded options. This would increase delta to 2,400 and require that 2,400
units of the asset be sold to maintain delta neutrality. The gamma of the
portfolio would change from —5,000 to —3,000.
To make the portfolio gamma and vega neutral, we suppose that there is a
second traded option with a gamma of 0.8, a vega of 1.2, and a delta of 0.5. If
w1 and w2 are the quantities of the two traded options included in the
portfolio, we require that
-5,000 + 0.5w1 + 0.8w2 = 0 and
- 8,000 + 2.0w1 + 1.2w2 = 0
The solution to these equations is w1 = 400, w2 = 6,000. The portfolio can
therefore be made gamma and vega neutral by including 400 of the first traded
option and 6,000 of the second traded option. The delta of the portfolio after
3
Vega is the name given to one of the "Greek letters" in option pricing, but it is not one
of the letters in the Greek alphabet.
How
Traders
Manage
Their
Exposures
67
Figure 3.6 Variation of vega of an option with price of
underlying asset. K is the option's strike price.
the addition of the positions in the two traded options is
400 x 0.6 + 6,000 x 0.5 = 3,240. Hence, 3,240 units of the asset would have
to be sold to maintain delta neutrality.
The vega of a long position in an option is positive. The variation of vega
with the price of the underlying asset is similar to that of gamma and is
shown in Figure 3.6. Gamma neutrality protects against large changes in
the price of the underlying asset between hedge rebalancing. Vega neutrality protects against variations in volatility.
The volatilities of short-dated options tend to be more variable than
the volatilities of long-dated options. The vega of a portfolio is therefore
often calculated by changing the volatilities of short-dated options by
more than that of long-dated options. This is discussed in Section 5.10.
3.4 THETA
The theta,
of a portfolio is the rate of change of the value of the
portfolio with respect to the passage of time with all else remaining the
same. Theta is sometimes referred to as the time decay of the portfolio.
Theta is usually negative for an option. 4 This is because as the time to
maturity decreases, with all else remaining the same, the option tends to
become less valuable. The general way in which varies with stock price
for a call option on a stock is shown in Figure 3.7. When the stock price is
very low, theta is close to zero. For an at-the-money call option, theta is
4
An exception to this could be an in-the-money European put option on a non-dividendPaying stock or an in-the-money European call option on a currency with a very high
interest rate.
68
Chapter 3
Figure 3.7
Variation of theta of a European call
option with stock price.
large and negative. Figure 3.8 shows typical patterns for the variation of
with the time to maturity for in-the-money, at-the-money, and out-ofthe-money call options.
Theta is not the same type of Greek letter as delta. There is uncertainty
about the future stock price, but there is no uncertainty about the passage
of time. It makes sense to hedge against changes in the price of the
Figure 3.8 Typical patterns for variation of theta of a European
call option with time to maturity.
How Traders Manage Their Exposures
69
underlying asset, but it does not make any sense to hedge against the
effect of the passage of time on an option portfolio. In spite of this, many
traders regard theta as a useful descriptive statistic for a portfolio. In a
delta-neutral portfolio, when theta is large and positive, gamma tends to
be large and negative, and vice versa.
3.5 RHO
The final Greek letter we consider is rho. Rho is the rate of change of a
portfolio with respect to the level of interest rates. Currency options have
two rhos, one for the domestic interest rate and one for the foreign interest
rate. When bonds and interest rate derivatives are part of a portfolio,
traders usually consider carefully the ways in which the whole term
structure of interest rates can change. We will discuss this in the next
chapter.
3.6 CALCULATING GREEK LETTERS
The calculation of Greek letters for options is explained in Appendices C
and D. The DerivaGem software, which can be downloaded from the
author's website, can be used to calculate Greek letters for both regular
options and exotics.
Consider again the European call option considered in Section 3.1.
The stock price is $49, the strike price is $50, the risk-free rate is 5%, the
stock price volatility is 20%, and the time to exercise is 20 weeks or 20/
52 years. Using the Analytic (European) calculation, we see that the
option price is $2.40; the delta is 0.522 (per $); the gamma is 0.066 (per $
per $); the vega is 0.121 per %; the theta is -0.012 per day; and the rho
is 0.089 per %.
These numbers imply the following:
1. When there is an increase of $0.10 in the stock price with no other
changes, the option price increases by about 0.522 x 0.1, or $0.0522.
2. When there is an increase $0.10 in the stock price with no other
changes, the delta of the option increases by about 0.066 x 0.1, or
0.0066.
3. When there is an increase of 0.5% in volatility with no other
changes, the option price increases by about 0.121 x 0.5, or 0.0605.
70
Chapter 3
4. When one day goes by with no changes to the stock price or its
volatility, the option price decreases by about 0.012.
5. When interest rates increase by 1% (or 100 basis points) with no
other changes, the option price increases by 0.089.
3.7 TAYLOR SERIES EXPANSIONS
A Taylor series expansion of the change in the portfolio value in a short
period of time shows the role played by different Greek letters. Consider a
portfolio dependent on a single market variable, S. If the volatility of the
underlying asset and interest rates are assumed to be constant, the value
of the portfolio,
is a function of S and time t. The Taylor series
expansion gives
(3.1)
where
and
are the change in and S, respectively, in a small time
interval
Delta hedging eliminates the first term on the right-hand side.
The second term, which is theta times
is nonstochastic. The third term
can be made zero by ensuring that the portfolio is gamma neutral as well
as delta neutral. Arguments from stochastic calculus show that
is of
order
This means that the third term on the right-hand side is of
order
Later terms in the Taylor series expansion are of higher order
than
For a delta-neutral portfolio, the first term on the right-hand side of
equation (3.1) is zero, so that
(3.2)
when terms of higher order than
are ignored. The relationship between
the change in the portfolio value and the change in the stock price is
quadratic as shown in Figure 3.9. When gamma is positive, the holder of
the portfolio gains from large movements in the market variable and loses
when there is little or no movement. When gamma is negative, the reverse
is true and a large positive or negative movement in the market variable
leads to severe losses.
When the volatility of the underlying asset is uncertain, is a function
How
Traders Manage
Their Exposures
71
(a)
(b)
(c)
(d)
Figure 3.9 Alternative relationships between
and
f or a deltaneutral portfolio, with (a) slightly positive gamma, (b) large positive
gamma, (c) slightly negative gamma, and (d) large negative gamma.
of
S, and t. Equation (3.1) then becomes
where
is the change in
in time
In this case, delta hedging
eliminates the first term on the right-hand side. The second term is
eliminated by making the portfolio vega neutral. The third term is
nonstochastic. The fourth term is eliminated by making the portfolio
gamma neutral.
Traders often define other "Greek letters" to correspond to higherorder terms in the Taylor series expansion. For example,
is
sometimes referred to as "gamma of vega".
Chapter
72
Business Snapshot 3.2
Dynamic Hedging in Practice
In a typical arrangement at a financial institution, the responsibility for a
portfolio of derivatives dependent on a particular underlying asset is assigned
to one trader or to a group of traders working together. For example, one
trader at Goldman Sachs might be assigned responsibility for all derivatives
dependent on the value of the Australian dollar. A computer system calculates
the value of the portfolio and Greek letters for the portfolio. Limits are
defined for each Greek letter and special permission is required if a trader
wants to exceed a limit at the end of a trading day.
The delta limit is often expressed as the equivalent maximum position in the
underlying asset. For example, the delta limit of Goldman Sachs on Microsoft
might be S10 million. If the Microsoft slock price is S50, this means that the
absolute value of delta as we have calculated it can be no more that 200,000.
The vega limit is usually expressed as a maximum dollar exposure per 1%
change in the volatility.
As a matter of course, options traders make themselves delta neutral -or
close to delta neutral at the end of each day. Gamma and vega are monitored, but arc not usually managed on a daily basis. Financial institutions
often find that their business with clients involves writing options and that as a
result they accumulate negative gamma and vega. They are then always
looking out for opportunities to manage their gamma and vega risks by
buying options at competitive prices.
There is one aspect of an options portfolio that mitigates problems of
managing gamma and vega somewhat. Options are often close to the money
when they are first sold so that they have relatively high gammas and vegas.
However, after some lime has elapsed, the underlying asset price has often
changed sufficiently for them to become deep out of the money or deep in the
money. Their gammas and vegas are then very small and of little consequence.
The nightmare scenario for an options trader is where written options remain
very close to the money as the maturity date is approached.
3.8 THE REALITIES OF HEDGING
In an ideal world traders working for financial institutions would be able
to rebalance their portfolios very frequently in order to maintain a zero
delta, a zero gamma, a zero vega, and so on. In practice, this is not
possible. When managing a large portfolio dependent on a single underlying asset, traders usually make delta zero, or close to zero, at least once
a day by trading the underlying asset. Unfortunately, a zero gamma and a
zero vega are less easy to achieve because it is difficult to find options or
How Traders Manage Their Exposures
Business Snapshot 3.3
73
Is Delta Hedging Easier or More Difficult
for Exotics?
We can approach the hedging of exotic options by creating a delta-neutral
position and rebalancing frequently to maintain delta neutrality. When we do
this, we find that some exotic options are easier to hedge than plain vanilla
options and some are more difficult.
An example of an exotic option that is relatively easy to hedge is an average
price call option (sec Asian options in Section 2.5). As time passes, we observe
more of the asset prices that will be used in calculating the final average. This
means that our uncertainty about the payoff decreases with the passage of
time. As a result, the option becomes progressively easier to hedge. In the final
few days, the delta of the option always approaches zero because price movements during this time have very little impact on the payoff.
By contrast, barrier options (see Section 2.5) are relatively difficult to hedge.
Consider a knock-out call option on a currency when the exchange rate is
0.0005 above the barrier. If the barrier is hit. the option is worth nothing. If it is
not hit. the option may prove to be quite valuable. The delta of the option is
discontinuous at the barrier, making conventional hedging very difficult.
other nonlinear derivatives that can be traded in the volume required at
competitive prices (see the discussion of dynamic hedging in Business
Snapshot 3.2).
There are large economies of scale in being an options trader. As noted
earlier, maintaining delta neutrality for an individual option on an asset
by trading the asset daily would be prohibitively expensive. But it is
realistic to do this for a portfolio of several hundred options on the asset.
This is because the cost of daily rebalancing is covered by the profit on
many different trades.
3.9 HEDGING EXOTICS
Exotic options can often be hedged using the approach we have outlined.
As explained in Business Snapshot 3.3, delta hedging is sometimes easier
for exotics and sometimes more difficult. When delta hedging is not
feasible for a portfolio of exotic options, an alternative approach known
as static options replication is sometimes used. This is illustrated in
Figure 3.10. Suppose that S denotes the asset price and t denotes time
With the current (t = 0) value of S being S0. Static options replication
involves choosing a barrier in {S, t}-space that will eventually be reached
74
Chapter 3
Value of exotic option
portfolio and portfolio П
is the same at these points
Figure 3.10 Static options replication. A replicating portfolio is
chosen so that it has the same value as the exotic option portfolio at a
number of points on a barrier.
and then finding a portfolio
of plain vanilla options that is worth the
same as the portfolio of exotic options at a number of points on the
barrier. The portfolio of exotic options is hedged by shorting
Once the
barrier is reached the hedge is unwound.
The theory underlying static options replication is that if two portfolios
are worth the same at all {S, t} points on the barrier they must be worth
the same at all the {S, t} points that can be reached prior to the barrier. In
practice, values of the original portfolio and the replicating portfolio
are matched at some, but not all, points on the barrier. The procedure
therefore relies on the idea that if two portfolios have the same value at a
reasonably large number of points on the barrier then their values are
likely to be close at other points on the barrier.
3.10 SCENARIO ANALYSIS
In addition to monitoring risks such as delta, gamma, and vega, option
traders often also carry out a scenario analysis. The analysis involves
How Traders Manage Their Exposures
Table 3.4
Profit or loss realized in two weeks under different
scenarios ($ millions)
Exchange rate
Volatility
8%
10%
12%
75
0.94
0.96
0.98
1.00
1.02
1.04
1.06
+102
+80
+60
+55
+40
+25
+25
+17
+6
+2
-2
-10
-14
-18
-34
-38
-42
-80
-85
-90
+9
calculating the gain or loss on their portfolio over a specified period
under a variety of different scenarios. The time period chosen is likely to
depend on the liquidity of the instruments. The scenarios can be either
chosen by management or generated by a model.
Consider a trader with a portfolio of options on a particular foreign
currency. There are two main variables on which the value of the portfolio depends. These are the exchange rate and the exchange rate volatility. Suppose that the exchange rate is currently 1.0000 and its volatility is
10% per annum. The bank could calculate a table such as Table 3.4
showing the profit or loss experienced during a two-week period under
different scenarios. This table considers seven different exchange rates
and three different volatilities. Because a one-standard-deviation move in
the exchange rate during a two-week period is usually about 0.02, the
exchange rate moves considered are approximately one, two, and three
standard deviations.
In Table 3.4 the greatest loss is in the lower right corner of the table.
The loss corresponds to the volatility increasing to 12% and the exchange
rate moving up to 1.06. Usually the greatest loss in a table such as 3.4
occurs at one of the corners, but this is not always so. For example, as we
saw in Figure 3.9, when gamma is positive the greatest loss is experienced
when the underlying market variable stays where it is.
SUMMARY
The individual responsible for the trades involving a particular market
variable monitors a number of Greek letters and ensures that they are
kept within the limits specified by his or her employer.
The delta, A, of a portfolio is the rate of change of its value with
respect to the price of the underlying asset. Delta hedging involves
76
Chapter 3
creating a position with zero delta (sometimes referred to as a deltaneutral position). Since the delta of the underlying asset is 1.0, one way of
hedging the portfolio is to take a position of —A in the underlying asset.
For portfolios involving options and more complex derivatives, the
position taken in the underlying asset has to be changed periodically.
This is known as rebalancing.
Once a portfolio has been made delta neutral, the next stage is often to
look at its gamma. The gamma of a portfolio is the rate of change of its
delta with respect to the price of the underlying asset. It is a measure of
the curvature of the relationship between the portfolio and the asset price.
Another important hedge statistic is vega. This measures the rate of
change of the value of the portfolio with respect to changes in the
volatility of the underlying asset. Gamma and vega can be changed by
trading options on the underlying asset.
In practice, derivatives traders usually rebalance their portfolios at least
once a day to maintain delta neutrality. It is usually not feasible to
maintain gamma and vega neutrality on a regular basis. Typically a
trader monitors these measures. If they get too large, either corrective
action is taken or trading is curtailed.
FURTHER READING
Derman, E., D. Ergener, and I. Kani, "Static Options Replication," Journal of
Derivatives, 2, No. 4 (Summer 1995), 78-95.
Taleb, N. N., Dynamic Hedging: Managing Vanilla and Exotic Options. New
York: Wiley, 1996.
QUESTIONS A N D PROBLEMS (Answers at End of Book)
3.1. The delta of a derivatives portfolio dependent on the S&P 500 index is
—2,100. The S&P 500 index is currently 1,000. Estimate what happens to
the value of the portfolio when the index increases to 1,005.
3.2. The vega of a derivatives portfolio dependent on the USD/GBP exchange
rate is 200 ($ per %). Estimate the effect on the portfolio of an increase in
the volatility of the exchange rate from 12% to 14%.
3.3. The gamma of a delta-neutral portfolio is 30 (per $ per $). Estimate what
happens to the value of the portfolio when the price of the underlying asset
(a) suddenly increases by $2 and (b) suddenly decreases by $2.
How Traders Manage Their Exposures
77
3.4. What does it mean to assert that the delta of a call option is 0.7? How can
a short position in 1,000 options be made delta neutral when the delta of a
long position in each option is 0.7?
3.5. What does it mean to assert that the theta of an option position is -100
per day? If a trader feels that neither a stock price nor its implied volatility
will change, what type of option position is appropriate?
3.6. What is meant by the gamma of an option position? What are the risks in
the situation where the gamma of a position is large and negative and the
delta is zero?
3.7. "The procedure for creating an option position synthetically is the reverse
of the procedure for hedging the option position." Explain this statement.
3.8. A company uses delta hedging to hedge a portfolio of long positions in put
and call options on a currency. Which of the following would lead to the
most favorable result: (a) a virtually constant spot rate or (b) wild movements in the spot rate? How does your answer change if the portfolio
contains short option positions?
3.9. A bank's position in options on the USD/euro exchange rate has a delta of
30,000 and a gamma of —80,000. Explain how these numbers can be
interpreted. The exchange rate (dollars per euro) is 0.90. What position
would you take to make the position delta neutral? After a short period of
time, the exchange rate moves to 0.93. Estimate the new delta. What
additional trade is necessary to keep the position delta neutral? Assuming
the bank did set up a delta-neutral position originally, has it gained or lost
money from the exchange rate movement?
3.10. "Static options replication assumes that the volatility of the underlying
asset will be constant." Explain this statement.
3.11. Suppose that a trader using the static options replication technique wants
to match the value of a portfolio of exotic derivatives with the value of a
portfolio of regular options at 10 points on a boundary. How many
regular options are likely to be needed? Explain your answer.
3.12. Why is an Asian option easier to hedge than a regular option?
3.13. Explain why there are economies of scale in hedging options.
3.14. Consider a six-month American put option on a foreign currency when the
exchange rate (domestic currency per foreign currency) is 0.75, the strike
price is 0.74, the domestic risk-free rate is 5%, the foreign risk-free rate is
3%, and the exchange rate volatility is 14% per annum. Use the DerivaGem software (binomial tree with 100 steps) to calculate the price, delta,
gamma, vega, theta, and rho of the option. (The software can be downloaded from the author's website.) Verify that delta is correct by changing
the exchange rate to 0.751 and recomputing the option price.
Chapter 3
78
ASSIGNMENT QUESTIONS
3.15. The gamma and vega of a delta-neutral portfolio are 50 per $ per $ and 25
per %, respectively. Estimate what happens to the value of the portfolio
when there is a shock to the market causing the underlying asset price to
decrease by $3 and its volatility to increase by 4%.
3.16. Consider a one-year European call option on a stock when the stock price
is $30, the strike price is $30, the risk-free rate is 5%, and the volatility is
25% per annum. Use the DerivaGem software to calculate the price, delta,
gamma, vega, theta, and rho of the option. Verify that delta is correct by
changing the stock price to $30.1 and recomputing the option price. Verify
that gamma is correct by recomputing the delta for the situation where the
stock price is $30.1. Carry out similar calculations to verify that vega,
theta, and rho are correct.
3.17. A financial institution has the following portfolio of over-the-counter
options on sterling:
Type
Position
Delta of
option
Gamma of
option
Vega of
option
Call
Call
Put
Call
-1000
-500
-2000
-500
0.50
0.80
-0.40
0.70
2.2
0.6
1.3
1.8
1.8
0.2
0.7
1.4
A traded option is available with a delta of 0.6, a gamma of 1.5, and a vega
of 0.8. (a) What position in the traded option and in sterling would make
the portfolio both gamma neutral and delta neutral? (b) What position in
the traded option and in sterling would make the portfolio both vega
neutral and delta neutral?
3.18. Consider again the situation in Problem 3.17. Suppose that a second
traded option with a delta of 0.1, a gamma of 0.5, and a vega of 0.6 is
available. How could the portfolio be made delta, gamma, and vega
neutral?
3.19. Reproduce Table 3.2. (In Table 3.2 the stock position is rounded to the
nearest 100 shares.) Calculate the gamma and theta of the position each
week. Calculate the change in the value of the portfolio each week (before
the rebalancing at the end of the week) and check whether equation (3.2) is
approximately satisfied. {Note: DerivaGem produces a value of theta "per
calendar day". The theta in the formula in Appendix C is "per year".)
Interest Rate Risk
Interest rate risk is more difficult to manage than the risk arising from
market variables such as equity prices, exchange rates, and commodity
prices. One complication is that there are many different interest rates in
any given currency (Treasury rates, interbank borrowing and lending
rates, mortgage rates, deposit rates, prime borrowing rates? and so on).
Although these tend to move together, they are not perfectly correlated.
Another complication is that, to describe an interest rate, we need more
than a single number. We need a function describing the variation of the
rate with maturity. This is known as the interest rate term structure or the
yield curve.
Consider, for example, the situation of a US government bond trader.
The trader's portfolio is likely to consist of many bonds with different
maturities. The trader has an exposure to movements in the one-year rate,
the two-year rate, the three-year rate, and so on. The trader's delta
exposure is therefore more complicated than that of the gold trader in
Table 3.1. The trader must be concerned with all the different ways in
which the US Treasury yield curve can change its shape through time.
This chapter starts with some preliminary material on types of interest
rates and the way interest rates are measured. It then moves on to consider
the ways exposures to interest rates can be managed. Duration and
convexity measures are covered first. For parallel shifts in the yield curve,
these are analogous to the delta and gamma measures discussed in the
previous chapter. A number of different approaches to managing the risks
Chapter 4
80
of nonparallel shifts are then presented. These include the use of partial
durations, the calculation of multiple deltas, and the use of principal
components analysis.
4.1 MEASURING INTEREST RATES
A statement by a bank that the interest rate on one-year deposits is 10%
per annum sounds straightforward and unambiguous. In fact, its precise
meaning depends on the way the interest rate is measured.
If the interest rate is measured with annual compounding, the bank's
statement that the interest rate is 10% means that $100 grows to
$100 x 1.1 =$110
at the end of one year. When the interest rate is measured with semiannual compounding, it means that we earn 5% every six months, with
the interest being reinvested. In this case, $100 grows to
$100 x 1.05 x 1.05 = $110.25
at the end of one year. When the interest rate is measured with quarterly
compounding, the bank's statement means that we earn 2.5% every three
months, with the interest being reinvested. The $100 then grows to
$100 x 1.0254 = $110.38
at the end of one year. Table 4.1 shows the effect of increasing the
compounding frequency further.
Table 4.1 Effect of the compounding frequency
on the value of $100 at the end of one year when
the interest rate is 10% per annum.
Compounding
frequency
Value of $100
at end of year {$)
Annually (m = 1)
Semiannually (m = 2)
Quarterly (m = 4)
Monthly (m = 12)
Weekly (m = 52)
Daily (m = 365)
110.00
110.25
110.38
110.47
110.51
110.52
Interest
81
Rate Risk
The compounding frequency defines the units in which an interest rate
is measured. A rate expressed with one compounding frequency can be
converted into an equivalent rate with a different compounding frequency. For example, from Table 4.1 we see that 10.25% with annual
compounding is equivalent to 10% with semiannual compounding. We
can think of the difference between one compounding frequency and
another to be analogous to the difference between kilometers and miles.
They are two different units of measurement.
To generalize our results, suppose that an amount A is invested for
n years at an interest rate of R per annum. If the interest is compounded
once per annum, the terminal value of the investment is
If the interest is compounded m times per annum, the terminal value of
the investment is
(4.1)
When m = 1, the rate is sometimes referred to as the equivalent annual
interest rate.
Continuous Compounding
The limit as the compounding frequency m tends to infinity is known as
continuous compounding.1 With continuous compounding, it can be
shown that an amount A invested for n years at rate R grows to
(4.2)
where e = 2.71828. The function ex is built into most calculators, so the
computation of the expression in equation (4.2) presents no problems. In
the example in Table 4.1, A = 100, n = 1, and R — 0.1, so that the value
to which A grows in one year with continuous compounding is
100e0.1 = $110.52
This is (to two decimal places) the same as the value with daily compounding. For most practical purposes, continuous compounding can be
thought of as being equivalent to daily compounding. Compounding a
sum of money at a continuously compounded rate R for n years involves
1
Actuaries sometimes refer to a continuously compounded rate as the force of interest.
82
Chapter 4
multiplying it by
Discounting it at a continuously compounded rate
R for n years involves multiplying by
Suppose that Rc is a rate of interest with continuous compounding and
Rm is the equivalent rate with compounding m times per annum. From
the results in equations (4.1) and (4.2), we must have
or
This means that
(4.3)
and
(4.4)
These equations can be used to convert a rate with a compounding
frequency of m times per annum to a continuously compounded rate
and vice versa. The function ln is the natural logarithm function and is
built into most calculators. This function is defined so that, if y = ln x,
then x = ey.
Example 4.1
Consider an interest rate that is quoted as 10% per annum with semiannual
compounding. From equation (4.3) with m = 2 and Rm =0.1, the equivalent
rate with continuous compounding is
or 9.758% per annum.
Example 4.2
Suppose that a lender quotes the interest rate on loans as 8% per annum with
continuous compounding and that interest is actually paid quarterly. From
equation (4.4) with m = 4 and Rc = 0.08, the equivalent rate with quarterly
compounding is
4(e 0.08/4 -1) = 0.0808
or 8.08% per annum. This means that on a $1,000 loan, interest payments of
$20.20 would be required each quarter:
Interest Rate Risk
83
4.2 ZERO RATES AND FORWARD RATES
The n-year zero-coupon interest rate is the rate of interest earned on an
investment that starts today and lasts for n years. All the interest and
principal is realized at the end of n years. There are no intermediate
payments. The n-year zero-coupon interest rate is sometimes also referred
to as the n-year spot rate, the n-year zero rate, or just the n-year zero. The
zero rate as a function of maturity is referred to as the zero curve. Suppose
a five-year zero rate with continuous compounding is 5% per annum.
This means that $100, if invested for five years, grows to
100 x e0.05x5 = 128.40
A forward rate is the future zero rate implied by today's zero rates.
Consider the zero rates shown in Table 4.2. The forward rate for the
period between six months and one year is 6.6%. This is because 5% for
the first six months combined with 6.6% for the next six months gives an
average of 5.8% for one year. Similarly, the forward rate for the period
between 12 months and 18 months is 7.6% because this rate when
combined with 5.8% for the first 12 months gives an average of 6.4%
for 18 months. In general, the forward rate F for the period between
times T1 and T2 is
(4.5)
where R1 is the zero rate for maturity of T1 and R2 is the zero rate for
maturity T2. This formula is exactly correct when rates are measured with
continuous compounding and approximately correct for other compounding frequencies. The results from using this formula on the rates
in Table 4.2 are given in Table 4.3. For example, substituting T1 = 1.5,
Table 4.2 Zero rates.
Maturity
(years)
Zero rate
(% cont. comp.)
0.5
1.0
1.5
2.0
5.0
5.8
6.4
6.8
Chapter 4
84
Table 4.3
Forward rates for zero rates
in Table 4.2.
Period
(years)
Forward rate
{% cont. comp.)
0.5 to 1.0
1.0 to 1.5
1.5 to 2.0
6.6
7.6
8.0
T2 = 2.0, R1 = 0.064, and R2 = 0.068, we get F = 0.08, showing that the
forward rate for the period between 18 months and 24 months is 8.0%.
Investors who think that future interest rates will be markedly different
from forward rates have no difficulty in finding trades that reflect their
beliefs (see Business Snapshot 4.1).
Bond Pricing
Most bonds provide coupons periodically. The bond's principal (also
known as its par value or face value) is received at the end of its life.
The theoretical price of a bond can be calculated as the present value of
all the cash flows that will be received by the owner of the bond. The most
accurate approach is to use a different zero rate for each cash flow. To
illustrate this, consider the situation where zero rates are as shown in
Table 4.2. Suppose that a two-year bond with a principal of $100 provides
coupons at the rate of 6% per annum semiannually. To calculate the
present value of the first coupon of $3, we discount it at 5.0% for six
months; to calculate the present value of the second coupon of $3, we
discount it at 5.8% for one year; and so on. The theoretical price of the
bond is therefore
or $98.39.
Bond Yields
A bond's yield is the discount rate that, when applied to all the bond's
cash flows, equates the bond price to its market price. Suppose that the
theoretical price of the bond we have been considering, $98.39, is also its
market value (i.e., the market's price of the bond is in exact agreement
with the data in Table 4.2). If y is the yield on the bond, expressed with
Interest Rate Risk
Business Snapshot 4.1
85
Orange County's Yield Curve Plays
Consider an investor who can borrow or lend at the rates shown in Table 4.2.
Suppose the investor thinks that the six-month interest rates will not change
much over the next three years. The investor can borrow six-month funds and
Invest for two years. The six-month borrowings can be rolled over at the end of
.6, 12. and 18 months. If interest rates do stay about the same, this strategy will
yield a profit of about 1.8% per year because interest will be received at 6.8%
and paid at 5%. This type of trading strategy is known as a yield curve play.
T|he investor is speculating that rates in the future will be quite different from
the forward rates shown in Table 4.3.
Robert Citron, the Treasurer at Orange County, used yield curve plays
similar to the one we have just described very successfully in 1992 and 1993.
The profit from Mr. Citron's trades became an important contributor to
Orange County's budget and he was re-elected. (No-one listened to his
opponent in the election, who said his trading strategy was too risky.)
In 1994 Mr. Citron expanded his yield curve plays. He invested
Heavily in inverse floaters. These pay a rate of interest equal to a fixed
rate of interest minus a floating rate. He also leveraged his position by
Borrowing at short-term interest rates. If short-term interest rates had
remained the same or declined, he would have continued to do well. As
it happened, interest rates rose sharply during 1994. On December 1,
19$94, Orange County announced that its investment portfolio had lost
$1.5 billion and several days later it filed for bankruptcy protection.
continuous compounding, we must have
This equation can be solved using Excel's Solver or in some other way to
give y = 6.76%.
4.3 TREASURY RATES
Treasury rates are the rates an investor earns on Treasury bills and
Treasury bonds. These are the instruments used by a government to
borrow in its own currency. Japanese Treasury rates are the rates at which
the Japanese government borrows in yen; US Treasury rates are the rates
at which the US government borrows in US dollars; and so on. It is
Usually assumed that there is no chance that a government will default on
86
Chapter 4
an obligation denominated in its own currency.2 Treasury rates are therefore totally risk-free rates in the sense that an investor who buys a
Treasury bill or Treasury bond is certain that interest and principal
payments will be made as promised.
Determining Treasury Zero Rates
One way of determining Treasury zero rates such as those in Table 4.2 is
to observe the yields on "strips". These are zero-coupon bonds that are
synthetically created by traders when they sell the coupons on a Treasury
bond separately from the principal.
Another way of determining Treasury zero rates is from regular Treasury bills and bonds. The most popular approach is known as the bootstrap method. This involves working from short maturities to successively
longer maturities and matching prices. Suppose that Table 4.2 gives the
Treasury rates determined so far and that a 2.5-year bond providing a
coupon of 8% sells for $102 per $100 of principal. We would determine
the 2.5-year zero rate as the rate R which, when used in conjunction with
the rates in Table 4.2, gives the correct price for this bond. This involves
solving
which gives R = 7.05%. The complete set of zero rates is shown in
Table 4.4. The zero curve is usually assumed to be linear between the
points that are determined by the bootstrap method. (In our example, the
2.25-year zero rate would be 6.925%.) It is also assumed to be constant
Table 4.4 Rates in Table 4.2 after 2.5-year rate
has been determined using the bootstrap method.
2
Maturity
(years)
Zero rate
(% cont. comp.)
0.5
1.0
1.5
2.0
2.5
5.00
5.80
6.40
6.80
7.05
The reason for this is that the government can always meet its obligation by printing
more money.
Interest Rate Risk
87
Zero rate (%)
Maturity (years)
Figure 4.1
Zero curve for data in Table 4.4.
prior to the first point and beyond the last point. The zero curve for our
example is shown in Figure 4.1.
4.4 LIBOR AND SWAP RATES
LIBOR is short for London Interbank Offered Rate. A LIBOR quote by a
particular bank is the rate of interest at which the bank is prepared to
make a large wholesale deposit with another bank. 3 Large banks and
other financial institutions quote 1-month, 3-month, 6-month, and
12-month LIBOR in all major currencies, where 1-month LIBOR is the
rate at which one-month deposits are offered, 3-month LIBOR is the rate
at which three-month deposits are offered, and so on. A deposit with a
bank can be regarded as a loan to that bank. A bank must therefore
satisfy certain creditworthiness criteria to qualify for receiving LIBOR
deposits. Typically, it must have an AA credit rating.4
3
Banks also quote LIBID, the London Interbank Bid Rate. The is the rate at which a
bank is prepared to accept deposits from another bank. The LIBOR quote is slightly
higher than the LIBID quote.
The best credit rating given to a company by the rating agency S&P is AAA. The
second best is AA. The corresponding ratings from the rival rating agency Moody's are
Aaa and Aa, respectively. More details on ratings are in Chapter 11.
88
Chapter 4
LIBOR rates are therefore the 1-month to 12-month borrowing rates
for banks (and other companies) that have AA credit ratings. How can
the LIBOR yield curve be extended beyond one year? There are two ways
of doing this:
1. Create a yield curve to represent the rates at which AA-rated
companies can borrow for periods of time longer than one year.
2. Create a yield curve to represent the future short-term borrowing
rates for AA-rated companies.
It is important to understand the difference. Suppose that the yield curve
is 4% for all maturities. If the yield curve is created in the first way, this
means that AA-rated companies can today lock in an interest rate of 4%
regardless of how long they want to borrow. If the yield curve is created in
the second way, then the forward interest rate that the market assigns to
the short-term borrowing rates of AA-rated companies at future times is
4%. When the yield curve is created in the first way, it gives the forward
short-term borrowing rate for a company that is AA-rated today. When it
is created in the second way, it gives the forward short-term borrowing
rate for a company that will be AA at the beginning of the period covered
by the forward contract.
In practice, the LIBOR yield curve is extended using the second
approach. The LIBOR yield curve is sometimes also called the swap yield
curve or the LIBOR/swap yield curve. The LIBOR/swap zero rates out to
one year are known directly from quoted LIBOR deposit rates. Swap rates
(see Table 2.5) allow the yield curve to be extended beyond one year using
an approach similar to the bootstrap method described for Treasuries in
the previous section.5 To understand why this is so, consider a bank that
1. Lends a certain principal for six months to an AA borrower and
relends it for successive six month periods to other AA borrowers,
and
2. Enters into a swap to exchange the LIBOR for the five-year swap
rate
These transactions show that the effective interest rate earned from the
series of short-term loans to AA borrowers is equivalent to the swap rate.
This means that the swap yield curve and the LIBOR yield curve (defined
using the second approach above) are the same.
5
Eurodollar futures, which are contracts on the future value of LIBOR, can also be used
to extend the LIBOR yield curve.
Interest
Rate Risk
89
The Risk-Free Rate
The risk-free rate is important in the pricing of financial contracts. The
usual practice among financial institutions is to assume that the LIBOR/
swap yield curve provides the risk-free rate. Treasury rates are regarded as
too low to be used as risk-free rates because:
1. Treasury bills and Treasury bonds must be purchased by financial
institutions to fulfill a variety of regulatory requirements. This
increases demand for these Treasury instruments driving their prices
up and their yields down.
2. The amount of capital a bank is required to hold to support an
investment in Treasury bills and bonds is substantially smaller than
the capital required to support a similar investment in other very
low-risk instruments.
3. In the United States, Treasury instruments are given a favorable tax
treatment compared with most other fixed-income investments
because they are not taxed at the state level.
As we have seen, the credit risk in the LIBOR/swap yield curve corresponds to the credit risk in a series of short-term loans to AA-rated
borrowers. It is therefore not totally risk free. There is a small chance
that an AA borrower will default during the life of a short-term loan. But
the LIBOR/swap yield curve is close to risk free and is widely used by
traders as a proxy for the risk-free yield curve. There is some evidence that
a true risk-free yield curve, uninfluenced by the factors affecting Treasury
rates that we have just mentioned, is about 10 basis points (= 0.1%) below
the LIBOR/swap yield curve.6 By contrast, Treasury rates are about
50 basis points (0.5%) below LIBOR/swap rates on average.
4.5 DURATION
Duration is a widely used measure of a portfolio's exposure to yield curve
movements. As its name implies, the duration of an instrument is a
measure of how long, on average, the holder of the instrument has to
wait before receiving cash payments. A zero-coupon bond that lasts n
years has a duration of n years. However, a coupon-bearing bond lasting
6
See J. Hull, M. Predescu, and A. White, "The Relationship Between Credit Default
Swap Spreads, Bond Yields, and Credit Rating Announcements," Journal of Banking and
Finance, 28 (November 2004), 2789-2811.
90
Chapter 4
n years has a duration of less than n years, because the holder receives
some of the cash payments prior to year n.
Suppose that a bond provides the holder with cash flows at time
for i = 1 , . . . , n. The price B and yield y (continuously compounded) are
related by
(4.6)
The duration D of the bond is defined as
(4.7)
This can be written as
(4.8)
The term in parentheses is the ratio of the present value of the cash flow at
time
to the bond price. The bond price is the present value of all
payments. The duration is therefore a weighted average of the times when
payments are made, with the weight applied to time being equal to the
proportion of the bond's total present value provided by the cash flow at
time . The sum of the weights is 1.0.
When a small change
in the yield is considered, it is approximately
true that
(4.9)
From equation (4.6), this becomes
(4.10)
(Note that there is an inverse relationship between B and y. When bond
yields increase, bond prices decrease; and when bond yields decrease,
bond prices increase.) From equations (4.7) and (4.10), we obtain the key
duration relationship
(4.11)
This can be written as
(4.12)
Interest Rate Risk
91
Table 4.5
Calculation of duration.
Time
(years)
Cash flow
($)
Present value
($)
Weight
Time x Weight
0.5
1.0
1.5
2.0
2.5
3.0
5
5
5
5
5
105
4.709
4.435
4.176
3.933
3.704
73.256
0.050
0.047
0.044
0.042
0.039
0.778
0.025
0.047
0.066
0.083
0.098
2.333
Total
130
94.213
1.000
2.653
Equation (4.12) is an approximate relationship between percentage
changes in a bond price and changes in its yield. The equation is easy
to use and is the reason why duration, first suggested by Macaulay in
1938, has become such a popular measure.
Consider a three-year 10% coupon bond with a face value of $100.
Suppose that the yield on the bond is 12% per annum with continuous
compounding. This means that y = 0.12. Coupon payments of $5 are
made every six months. Table 4.5 shows the calculations necessary to
determine the bond's duration. The present values of the bond's cash
flows, using the yield as the discount rate, are shown in column 3. (For
example, the present value of the first cash flow is 5e-0.12x0.5 — 4.709.)
The sum of the numbers in column 3 gives the bond's price as 94.213. The
weights are calculated by dividing the numbers in column 3 by 94.213.
The sum of the numbers in column 5 gives the duration as 2.653 years.
Small changes in interest rates are often measured in basis points. A
basis point is 0.01% per annum. The following example investigates the
accuracy of the duration relationship in equation (4.11).
Example 4.5
For the bond in Table 4.5, the bond price B is 94.213 and the duration D is
2.653, so that equation (4.11) gives
or
When the yield on the bond increases by 10 basis points (=0.1%),
= +0.001. The duration relationship predicts that
92
Chapter 4
so that the bond price goes down to 94.213 — 0.250 = 93.963. How accurate is
this? When the bond yield increases by 10 basis points to 12.1%, the bond
price is
which is (to three decimal places) the same as that predicted by the duration
relationship.
Modified Duration
The preceding analysis is based on the assumption that y is expressed with
continuous compounding. If y is expressed with annual compounding, it
can be shown that the approximate relationship in equation (4.11)
becomes
More generally, if y is expressed with a compounding frequency of m
times per year, then
A variable D* defined by
is sometimes referred to as the bond's modified duration. It allows the
duration relationship to be simplified to
when y is expressed with a compounding frequency of m times per year.
The following example investigates the accuracy of the modified duration
relationship.
Example 4.6
The bond in Table 4.5 has a price of 94.213 and a duration of 2.653. The yield,
expressed with semiannual compounding is 12.3673%. The modified duration
D* is
From equation (4.13), we have
Interest Rate Risk
93
or
When the yield (semiannually compounded) increases by 10 basis points
(=0.1%),
=+0.001. The duration relationship predicts that we expect
to be —235.39 x 0.001 = —0.235, so that the bond price goes down to
94.213 - 0.235 = 93.978. How accurate is this? When the bond yield (semiannually compounded) increases by 10 basis points to 12.4673% (or to
12.0941% with continuous compounding), an exact calculation similar to that
in the previous example shows that the bond price becomes 93.978. This shows
that the modified duration calculation is accurate for small yield changes.
4.6 CONVEXITY
The duration relationship measures exposure to small changes in yields.
This is illustrated in Figure 4.2, which shows the relationship between the
percentage change in value and change in yield for bonds having the same
duration. The gradients of the two curves are the same at the origin. This
means that both portfolios change in value by the same percentage for
small yield changes, as predicted by equation (4.12). For large yield
Figure 4.2 Two portfolios with the same duration.
94
Chapter 4
changes, the portfolios behave differently. Portfolio X has more curvature
in its relationship with yields than Portfolio Y. A factor known as
convexity measures this curvature and can be used to improve the
relationship in equation (4.12).
A measure of convexity for a bond is
where y is the bond's yield. This is the weighted average of the square of
the time to the receipt of cash flows. From Taylor series expansions, a
more accurate expression than equation (4.9) is
This leads to
Example 4.7
Consider again the bond in Table 4.5: the bond price B is 94.213 and the
duration D is 2.653. The convexity is
0.05 x 0.52 + 0.047 x 1.02 + 0.044 x 1.52 + 0.042 x 2.02
+ 0.039 x 2.52 + 0.779 x 3.02 = 7.570
The convexity relationship in equation (4.14) is therefore
Consider a 2% change in the bond yield from 12% to 14%. The duration
relationship predicts that the dollar change in the value of the bond will be
—94.213 x 2.653 x 0.02 = —4.999. The convexity relationship predicts that it
will be
-94.213 x 2.653 x 0.02 + 0.5 x 94.213 x 7.570 x 0.022 = -4.856
The actual change in the value of the bond is —4.859. This shows that the
convexity relationship gives much more accurate results than duration for a
large change in the bond yield.
4.7 APPLICATION TO PORTFOLIOS
The duration concept can be used for any portfolio of assets dependent
on interest rates. Suppose that P is the value of the portfolio. We make a
Interest
Rate Risk
95
small parallel shift in the zero-coupon yield curve and observe the change
in P. Duration is defined as
where
is size of the parallel shift. Equation (4.12) becomes
Suppose the portfolio consists of a number of assets. The ith asset is
worth Xi and has a duration Di (i = 1 , . . . , n). Define
as the change
in the value of Xi arising from the yield curve shift
It follows that
so that the duration of the portfolio is
given by
The duration of the ith asset is
Hence,
This shows that the duration D of a portfolio is the weighted average of
the durations of the individual assets comprising the portfolio with the
weight assigned to an asset being proportional to the value of the asset.
The convexity can be generalized in the same way as the duration. For
an interest-rate-dependent portfolio with value P, we define the convexity
as 1/P times the second partial derivative of the value of the portfolio
with respect to a parallel shift in the zero-coupon yield curve. Equation
(4.14) is correct with B replaced by P:
(4.16)
The relationship between the convexity of a portfolio and the convexity of
the assets comprising the portfolio is similar to that for duration: the
convexity of the portfolio is the weighted average of the convexities of the
assets with the weights being proportional to the value of the assets.
96
Chapter 4
The convexity of a bond portfolio tends to be greatest when the
portfolio provides payments evenly over a long period of time. It is least
when the payments are concentrated around one particular point in time.
Portfolio Immunization
A portfolio consisting of long and short positions in interest-ratedependent assets can be protected against relatively small parallel shifts
in the yield curve by ensuring that its duration is zero. It can be
protected against relatively large parallel shifts in the yield curve by
ensuring that its duration and convexity are both zero or close to zero.
In this respect duration and convexity are analogous to the delta and
gamma Greek letters we encountered in Chapter 3.
4.8 NONPARALLEL YIELD CURVE SHIFTS
Unfortunately, the basic duration relationship in equation (4.15) only
quantifies exposure to parallel yield curve shifts. The duration plus
convexity relationship in equation (4.16) allows the shift to be relatively
large, but it is still a parallel shift.
Some researchers have attempted to extend duration measures so that
nonparallel shifts can be considered. Reitano suggests a partial duration
measure where just one point on the zero-coupon yield curve is shifted
and all other points remain the same.7 Suppose that the zero curve is as
shown in Table 4.6 and Figure 4.3. Shifting the five-year point involves
changing the zero curve as indicated in Figure 4.4. In general, the partial
duration of the portfolio for the ith point on the zero curve is
where
is the size of the small change made to the ith point on the yield
curve and
is the resultant change in the portfolio value. The sum of
all the partial duration measures equals the usual duration measure.
Suppose that the partial durations for a particular portfolio are as
shown in Table 4.7. The total duration of the portfolio is only 0.2. This
means that the portfolio is relatively insensitive to parallel shifts in the
yield curve. However, the durations for short maturities are positive while
7
See R. Reitano, "Non-Parallel Yield Curve Shifts and Immunization," Journal of
Portfolio Management, Spring 1992, 36-43.
Interest Rate Risk
Table 4.6
97
Zero-coupon yield curve (rates continuously compounded).
1
2
3
4
5
7
10
4.0
4.5
4.8
5.0
5.1
5.2
5.3
Maturity (years)
Rate (%)
Figure 4.3
Figure 4.4
The zero-coupon yield curve in Table 4.6.
Change in zero-coupon yield curve when one point is shifted.
Table 4.7
Maturity (years)
Duration
Partial durations for a portfolio.
1
2
3
4
5
7
10
Total
2.0
1.6
0.6
0.2
-0.5
-1.8
-1.9
0.2
98
Chapter 4
Figure 4.5 A rotation of the yield curve.
those for long maturities are negative. This means that the portfolio loses
(gains) in value when short rates rise (fall). It gains (loses) in value when
long rates rise (fall).
We are now in a position to go one step further and calculate the
impact of nonparallel shifts. We can define any type of shift we want.
Suppose that, in the case of the yield curve shown in Figure 4.3, we define
a rotation where the changes to the 1-year, 2-year, 3-year, 4-year, 5-year,
7-year, and 10-year points are — 3e, —2e, —e, 0, e, 3e, and 6e for some
small e. This is illustrated in Figure 4.5. From the partial durations in
Table 4.7, the percentage change in the value of the portfolio arising from
the rotation is
2.0 x (-3e) + 1.6 x (-2e) + 0.6 x (-e) + 0.2 x 0
- 0.5 x e - 1.8 x 3e - 1.9 x 6e = -27.1e
This shows that a portfolio that gives rise to the partial durations in Table
4.7 is much more heavily exposed to a rotation of the yield curve than to a
parallel shift.
4.9 INTEREST RATE DELTAS
We now move on to consider how the Greek letters discussed in Chapter 3
can be calculated for interest rates. One possibility is to define the delta of
a portfolio as the change in value for a one-basis-point parallel shift in
the zero curve. This is sometimes termed a DV01. It is the same as the
Interest Rate Risk
99
Table 4.8 Deltas for portfolio in Table 4.7. Value of Portfolio is $1 million.
The dollar impact of a one-basis-point shift in points on the zero curve is shown.
Maturity (years)
Delta
1
2
3
4
5
7
10
Total
200
160
60
20
-50
-180
-190
20
duration of the portfolio multiplied by the value of the portfolio multiplied by 0.0001.
In practice, analysts like to calculate several deltas to reflect their
exposures to all the different ways in which the yield curve can move.
There are a number of different ways this can be done. One approach
corresponds to the partial duration approach that we outlined in the
previous section. It involves computing the impact of a one-basis-point
change similar to the one illustrated in Figure 4.4 for each point on the
zero-coupon yield curve. This delta is the partial duration calculated in
Table 4.7 multiplied by the value of the portfolio multiplied by 0.0001.
The sum of the deltas for all the points on the yield curve equals the
DV01. Suppose that the portfolio in Table 4.7 is worth $1 million. The
deltas are shown in Table 4.8.
A variation on this approach is to divide the yield curve into a number
of segments or "buckets" and calculate for each bucket the impact of
changing all the zero rates corresponding to the bucket by one basis
point while keeping all other zero rates unchanged. This approach is
often used in asset-liability management (see Section 1.5) and is referred
to as GAP management. Figure 4.6 shows the type of change that would
Figure 4.6 Change considered to yield curve when bucketing approach is used.
100
Chapter 4
be considered for the segment of the zero curve between 2.0 and 3.0 years
in Figure 4.3. Again, the sum of the deltas for all the segments equals
the DV01.
Calculating Deltas to Facilitate Hedging
One of the problems with the delta measures that we have considered so
far is that they are not designed to make hedging easy. Consider the deltas
in Table 4.8. If we plan to hedge our portfolio with zero-coupon bonds,
we can calculate the position in a one-year zero-coupon bond to zero out
the $200 per basis point exposure to the one-year rate, the position in a
two-year zero-coupon bond to zero out the exposure to the two-year rate,
and so on. But, if other instruments are used, a much more complicated
analysis is necessary.
In practice, traders tend to use positions in the instruments that have
been used to construct the zero curve to hedge their exposure. For
example, a government bond trader is likely to take positions in the
actively traded government bonds that were used to construct the Treasury zero curve when hedging. A trader of instruments dependent on the
LIBOR/swap yield curve is likely to take positions in LIBOR deposits,
Eurodollar futures, and swaps when hedging.
To facilitate hedging, traders therefore often calculate the impact of
small changes in the quotes for each of the instruments used to construct
the zero curve. Consider a trader responsible for interest rate caps and
swap options. Suppose that the trader's exposure to a one-basis-point
change in a Eurodollar futures quote is $500. Each Eurodollar futures
contract changes in value by $25 for a one-basis-point change in the
Eurodollar futures quote. It follows that the trader's exposure can be
hedged with 20 contracts. Suppose that the exposure to a one-basis-point
change in the five-year swap rate is $4,000 and that a five-year swap with a
notional principal of $ 1 million changes in value by $400 for a one-basispoint change in the five-year swap rate. The exposure can be hedged by
trading swaps with a notional principal of $10 million.
4.10 PRINCIPAL COMPONENTS ANALYSIS
The approaches we have just outlined can lead to analysts calculating
10 to 15 different deltas for every zero curve. This seems like overkill
because the variables being considered are quite highly correlated with
each other. For example, when the yield on a five-year bond moves up by
Interest
Rate
Risk
101
Table 4 . 9
3m
6m
12m
2y
3y
4y
5y
7y
10y
30y
Factor loadings for US Treasury data.
PC1
PC2
PC3
PC4
PC5
PC6
PC7
PC8
PC9
PC10
0.21
0.26
0.32
0.35
0.36
0.36
0.36
0.34
0.31
0.25
-0.57
-0.49
-0.32
-0.10
0.02
0.14
0.17
0.27
0.30
0.33
0.50
0.23
-0.37
-0.38
-0.30
-0.12
-0.04
0.15
0.28
0.46
0.47
-0.37
-0.58
0.17
0.27
0.25
0.14
0.01
-0.10
-0.34
-0.39
0.70
-0.52
0.04
0.07
0.16
0.08
0.00
-0.06
-0.18
-0.02
0.01
-0.23
0.59
0.24
-0.63
-0.10
-0.12
0.01
0.33
0.01
-0.04
-0.04
0.56
-0.79
0.15
0.09
0.13
0.03
-0.09
0.00
-0.02
-0.05
0.12
0.00
0.55
-0.26
-0.54
-0.23
0.52
0.01
-0.01
0.00
-0.12
-0.09
-0.14
0.71
0.00
-0.63
0.26
0.00
0.00
0.01
-0.05
-0.00
-0.08
0.48
-0.68
0.52
-0.13
a few basis points, most of the time the yield on a ten-year bond does the
same. Arguably a trader should not be worried when a portfolio has a
large positive exposure to the five-year rate and a similar large negative
exposure to the ten-year rate.
One approach to handling the risk arising from groups of highly
correlated market variables is principal components analysis. This takes
historical data on movements in the market variables and attempts to
define a set of components or factors that explain the movements.
The approach is best illustrated with an example. The market variables
we will consider are ten US Treasury rates with maturities between three
months and 30 years. Tables 4.9 and 4.10 show results produced by Frye
for these market variables using 1,543 daily observations between 1989
and 1995.8 The first column in Table 4.9 shows the maturities of the rates
that were considered. The remaining ten columns in the table show the
ten factors (or principal components) describing the rate moves. The first
factor, shown in the column labeled PC1, corresponds to a roughly
parallel shift in the yield curve. When we have one unit of that factor,
Table 4.10
Standard deviation of factor scores (basis points).
PCI
PC2
PC3
PC4
PC5
PC6
PC7
PC8
PC9
PC10
17.49
6.05
3.10
2.17
1.97
1.69
1.27
1.24
0.80
0.79
8
See J. Frye, "Principals of Risk: Finding VAR through Factor-Based Interest Rate
Scenarios." In VAR: Understanding and Applying Value at Risk, Risk Publications,
London, 1997, pp. 275-288.
102
Chapter 4
the three-month rate increases by 0.21 basis points, the six-month rate
increases by 0.26 basis points, and so on. The second factor is shown in
the column labeled PC2. It corresponds to a "twist" or change of slope of
the yield curve. Rates between three months and two years move in one
direction; rates between three years and 30 years move in the other
direction. The third factor corresponds to a "bowing" of the yield curve.
Rates at the short end and long end of the yield curve move in one
direction; rates in the middle move in the other direction. The interest rate
move for a particular factor is known as factor loading. In our example,
the first factor's loading for the three-month rate is 0.21. 9
As there are ten rates and ten factors, the interest rate changes observed
on any given day can always be expressed as a linear sum of the factors by
solving a set of ten simultaneous equations. The quantity of a particular
factor in the interest rate changes on a particular day is known as the
factor score for that day.
The importance of a factor is measured by the standard deviation of its
factor score. The standard deviations of the factor scores in our example
are shown in Table 4.10 and the factors are listed in order of their
importance. The numbers in Table 4.10 are measured in basis points. A
quantity of the first factor equal to one standard deviation, therefore,
corresponds to the three-month rate moving by 0.21 x 17.49 = 3.67 basis
points, the six-month rate moving by 0.26 x 17.49 = 4.55 basis points,
and so on.
The technical details of how the factors are determined are not covered
here. It is sufficient for us to note that the factors are chosen so that the
factor scores are uncorrelated. For instance, in our example, the first
factor score (amount of parallel shift) is uncorrelated with the second
factor score (amount of twist) across the 1,543 days. The variances of the
factor scores (i.e., the squares of the standard deviations) have the
property that they add up to the total variance of the data. From
Table 4.10, the total variance of the original data (i.e., sum of the
variance of the observations on the three-month rate, the variance of
the observations on the six-month rate, and so on) is
17.492 + 6.052 + 3.102 + ... + 0.792 = 367.9
From this, it can be seen that the first factor accounts for
17.492/367.9 = 83.1% of the variance in the original data; the first two
factors account for (17.492 + 6.052)/367.9 = 93.1% of the variance in the
9
The factor loadings have the property that the sum of their squares for each factor is 1.0.
Interest Rate Risk
103
Figure 4.7 The three most important factors driving yield curve movements.
data; the third factor accounts for a further 2.6% of the variance. This
shows that most of the risk in interest rate moves is accounted for by the
first two or three factors. It suggests that we can relate the risks in a
portfolio of interest-rate-dependent instruments to movements in these
factors instead of considering all ten interest rates. The three most
important factors from Table 4.9 are plotted in Figure 4.7.10
Using Principal Components Analysis to Calculate Deltas
To illustrate how a principal components analysis can provide an alternative way of calculating deltas, suppose we have a portfolio with the
exposures to interest rate moves shown in Table 4.11. A one-basis-point
change in the one-year rate causes the portfolio value to increase by
$10 million; a one-basis-point change in the two-year rate causes it to
increase by $4 million; and so on. We use the first two factors to model
rate moves. (As mentioned earlier, this captures over 90% of the uncertainty in rate moves.) Using the data in Table 4.9, our delta exposure to
the first factor (measured in millions of dollars per factor-score basis
10
Results similar to those described here, with respect to the nature of the factors and
the amount of the total risk they account for, are obtained when a principal components
analysis is used to explain the movements in almost any yield curve in any country.
100
Chapter 4
be considered for the segment of the zero curve between 2.0 and 3.0 years
in Figure 4.3. Again, the sum of the deltas for all the segments equals
the DV01.
Calculating Deltas to Facilitate Hedging
One of the problems with the delta measures that we have considered so
far is that they are not designed to make hedging easy. Consider the deltas
in Table 4.8. If we plan to hedge our portfolio with zero-coupon bonds,
we can calculate the position in a one-year zero-coupon bond to zero out
the $200 per basis point exposure to the one-year rate, the position in a
two-year zero-coupon bond to zero out the exposure to the two-year rate,
and so on. But, if other instruments are used, a much more complicated
analysis is necessary.
In practice, traders tend to use positions in the instruments that have
been used to construct the zero curve to hedge their exposure. For
example, a government bond trader is likely to take positions in the
actively traded government bonds that were used to construct the Treasury zero curve when hedging. A trader of instruments dependent on the
LIBOR/swap yield curve is likely to take positions in LIBOR deposits,
Eurodollar futures, and swaps when hedging.
To facilitate hedging, traders therefore often calculate the impact of
small changes in the quotes for each of the instruments used to construct
the zero curve. Consider a trader responsible for interest rate caps and
swap options. Suppose that the trader's exposure to a one-basis-point
change in a Eurodollar futures quote is $500. Each Eurodollar futures
contract changes in value by $25 for a one-basis-point change in the
Eurodollar futures quote. It follows that the trader's exposure can be
hedged with 20 contracts. Suppose that the exposure to a one-basis-point
change in the five-year swap rate is $4,000 and that a five-year swap with a
notional principal of $1 million changes in value by $400 for a one-basispoint change in the five-year swap rate. The exposure can be hedged by
trading swaps with a notional principal of $10 million.
4.10 PRINCIPAL COMPONENTS ANALYSIS
The approaches we have just outlined can lead to analysts calculating
10 to 15 different deltas for every zero curve. This seems like overkill
because the variables being considered are quite highly correlated with
each other. For example, when the yield on a five-year bond moves up by
Interest Rate Risk
101
Table 4.9
3m
6m
12m
2y
3y
4y
5y
7y
l0y
30y
Factor loadings for US Treasury data.
PC1
PC2
PC3
PC4
PC5
PC6
PC7
PC8
PC9
PC10
0.21
0.26
0.32
0.35
0.36
0.36
0.36
0.34
0.31
0.25
-0.57
-0.49
-0.32
-0.10
0.02
0.14
0.17
0.27
0.30
0.33
0.50
0.23
-0.37
-0.38
-0.30
-0.12
-0.04
0.15
0.28
0.46
0.47
-0.37
-0.58
0.17
0.27
0.25
0.14
0.01
-0.10
-0.34
-0.39
0.70
-0.52
0.04
0.07
0.16
0.08
0.00
-0.06
-0.18
-0.02
0.01
-0.23
0.59
0.24
-0.63
-0.10
-0.12
0.01
0.33
0.01
-0.04
-0.04
0.56
-0.79
0.15
0.09
0.13
0.03
-0.09
0.00
-0.02
-0.05
0.12
0.00
0.55
-0.26
-0.54
-0.23
0.52
0.01
-0.01
0.00
-0.12
-0.09
-0.14
0.71
0.00
-0.63
0.26
0.00
0.00
0.01
-0.05
-0.00
-0.08
0.48
-0.68
0.52
-0.13
a few basis points, most of the time the yield on a ten-year bond does the
same. Arguably a trader should not be worried when a portfolio has a
large positive exposure to the five-year rate and a similar large negative
exposure to the ten-year rate.
One approach to handling the risk arising from groups of highly
correlated market variables is principal components analysis. This takes
historical data on movements in the market variables and attempts to
define a set of components or factors that explain the movements.
The approach is best illustrated with an example. The market variables
we will consider are ten US Treasury rates with maturities between three
months and 30 years. Tables 4.9 and 4.10 show results produced by Frye
for these market variables using 1,543 daily observations between 1989
and 1995.8 The first column in Table 4.9 shows the maturities of the rates
that were considered. The remaining ten columns in the table show the
ten factors (or principal components) describing the rate moves. The first
factor, shown in the column labeled PC1, corresponds to a roughly
Parallel shift in the yield curve. When we have one unit of that factor,
Table 4.10
Standard deviation of factor scores (basis points).
PC1
PC2
PC3
PC4
PC5
PC6
PC7
PC8
PC9
PC10
17.49
6.05
3.10
2.17
1.97
1.69
1.27
1.24
0.80
0.79
8
See J. Frye, "Principals of Risk: Finding VAR through Factor-Based Interest Rate
Scenarios." In VAR: Understanding and Applying Value at Risk, Risk Publications,
London, 1997, pp. 275-288.
102
Chapter 4
the three-month rate increases by 0.21 basis points, the six-month rate
increases by 0.26 basis points, and so on. The second factor is shown in
the column labeled PC2. It corresponds to a "twist" or change of slope of
the yield curve. Rates between three months and two years move in one
direction; rates between three years and 30 years move in the other
direction. The third factor corresponds to a "bowing" of the yield curve.
Rates at the short end and long end of the yield curve move in one
direction; rates in the middle move in the other direction. The interest rate
move for a particular factor is known as factor loading. In our example,
the first factor's loading for the three-month rate is 0.21. 9
As there are ten rates and ten factors, the interest rate changes observed
on any given day can always be expressed as a linear sum of the factors by
solving a set of ten simultaneous equations. The quantity of a particular
factor in the interest rate changes on a particular day is known as the
factor score for that day.
The importance of a factor is measured by the standard deviation of its
factor score. The standard deviations of the factor scores in our example
are shown in Table 4.10 and the factors are listed in order of their
importance. The numbers in Table 4.10 are measured in basis points. A
quantity of the first factor equal to one standard deviation, therefore,
corresponds to the three-month rate moving by 0.21 x 17.49 = 3.67 basis
points, the six-month rate moving by 0.26 x 17.49 = 4.55 basis points,
and so on.
The technical details of how the factors are determined are not covered
here. It is sufficient for us to note that the factors are chosen so that the
factor scores are uncorrelated. For instance, in our example, the first
factor score (amount of parallel shift) is uncorrelated with the second
factor score (amount of twist) across the 1,543 days. The variances of the
factor scores (i.e., the squares of the standard deviations) have the
property that they add up to the total variance of the data. From
Table 4.10, the total variance of the original data (i.e., sum of the
variance of the observations on the three-month rate, the variance of
the observations on the six-month rate, and so on) is
17.492 + 6.052 + 3.102 + • • • + 0.792 = 367.9
From this, it can be seen that the first factor accounts for
17.492/367.9 = 83.1% of the variance in the original data; the first two
factors account for (17.492 + 6.052)/367.9 = 93.1% of the variance in the
9
The factor loadings have the property that the sum of their squares for each factor is 1.0.
Interest Rate Risk
103
Figure 4.7 The three most important factors driving yield curve movements.
data; the third factor accounts for a further 2.6% of the variance. This
shows that most of the risk in interest rate moves is accounted for by the
first two or three factors. It suggests that we can relate the risks in a
portfolio of interest-rate-dependent instruments to movements in these
factors instead of considering all ten interest rates. The three most
important factors from Table 4.9 are plotted in Figure 4.7. 10
Using Principal Components Analysis to Calculate Deltas
To illustrate how a principal components analysis can provide an alternative way of calculating deltas, suppose we have a portfolio with the
exposures to interest rate moves shown in Table 4.11. A one-basis-point
change in the one-year rate causes the portfolio value to increase by
$10 million; a one-basis-point change in the two-year rate causes it to
increase by $4 million; and so on. We use the first two factors to model
rate moves. (As mentioned earlier, this captures over 90% of the uncertainty in rate moves.) Using the data in Table 4.9, our delta exposure to
the first factor (measured in millions of dollars per factor-score basis
10
Results similar to those described here, with respect to the nature of the factors and
the amount of the total risk they account for, are obtained when a principal components
analysis is used to explain the movements in almost any yield curve in any country.
Chapter 4
104
Table 4.11 Change in portfolio value for a
one-basis-point rate move ($ millions).
1-year
rate
2-year
rate
3-year
rate
4-year
rate
5-year
rate
+ 10
+4
-8
-1
+2
point) is
10 x 0.32 + 4 x 0.35 - 8 x 0.36 - 7 x 0.36 + 2 x 0.36 = -0.08
and our delta exposure to the second factor is
10 x (-0.32) + 4 x (-0.10) - 8 x 0.02 - 7 x 0.14 + 2 x 0.17 = -4.40
The approach being used here is similar to the approach described in
Section 4.8 where partial durations are used to estimate the impact of
nonparallel shifts. The advantage of using a principal components analysis is that it tells you which are the most appropriate shifts to consider.
It also provides information on the relative importance of different shifts.
In the example we have considered, our exposure to the second shift is
about 50 times greater than our exposure to the first shift. However, the
first shift is about three times as important in terms of the extent to which
it occurs. (We base this last statement on the standard deviation of factor
scores reported in Table 4.10.)
4.11 GAMMA AND VEGA
When several delta measures are calculated, there are many possible
gamma measures. Suppose that ten instruments are used to compute
the zero curve and that we measure deltas with respect to changes in
the quotes for each of these. Gamma is a second partial derivative of the
form
, where P is the portfolio value. We have ten choices for
xi and ten choices for xj and a total of 55 different gamma measures. This
may be "information overload". One approach is to ignore cross-gammas
and focus on the ten partial derivatives where i = j. Another is to
calculate a single gamma measure as the second partial derivative of
the value of the portfolio with respect to a parallel shift in the zero curve.
A further possibility is to calculate gammas with respect to the first two
factors in a principal components analysis.
,
Interest Rate Risk
105
The vega of a portfolio of interest rate derivatives measures its
exposure to volatility changes. Different volatilities are used to price
different interest rate derivatives. One approach is to make the same
small change to all volatilities and calculate the effect on the value of the
portfolio. Another is to carry out a principal components analysis to
calculate factors that reflect the patterns of volatility changes across
different instruments that tend to occur in practice. Vega measures can
be calculated for the first two or three factors.
SUMMARY
The compounding frequency used for an interest rate defines the units in
which it is measured. The difference between an annually compounded
rate and a quarterly compounded rate is analogous to the difference
between a distance measured in miles and a distance measured in kilometers. Analysts frequently use continuous compounding when analyzing
derivatives.
Many different types of interest rates are quoted in financial markets
and calculated by analysts. The n-year zero rate or n-year spot rate is the
rate applicable to an investment lasting for n years when all of the return
is realized at the end. Forward rates are the rates applicable to future
periods of time implied by today's zero rates.
A zero-coupon yield curve shows the zero rate as a function of
maturity. Two important zero-coupon yield curves for risk managers
are the Treasury zero curve and the LIBOR/swap zero curve. The method
most commonly used to calculate zero curves is known as the bootstrap
method. It involves starting with short-term instruments and moving to
progressively longer-term instruments making sure that the zero rates
calculated at each stage are consistent with the prices of the instruments.
An important concept in interest rate markets is duration. Duration
measures the sensitivity of the value of a portfolio to a small parallel shift
m the zero-coupon yield curve. An approximate relationship is
where P is the value of the portfolio, D is the duration of the portfolio, y
is the size of a small parallel shift in the zero curve, and P is the resultant
effect on the value of the portfolio. A more precise relationship is
106
Chapter 4
where C is the convexity of the portfolio. This relationship is accurate for
relatively large parallel shifts in the yield curve but does not quantify the
exposure to nonparallel shifts.
To quantify exposure to all the different ways the yield curve can
change through time, several duration or delta measures are necessary.
There are a number of ways these can be defined. A principal components
analysis can be a useful alternative to calculating multiple deltas. It shows
that the yield curve shifts that occur in practice are to a large extent a
linear sum of two or three standard shifts. If a portfolio manager is
hedged against these standard shifts, he or she is therefore also well
hedged against the shifts that occur in practice.
FURTHER READING
Allen, S. L., and A. D. Kleinstein. Valuing Fixed-Income Investments and
Derivative Securities. New York Institute of Finance, 1991.
Duffie, D. "Debt Management and Interest Rate Risk," in W. Beaver and
G. Parker (eds.) Risk Management: Challenges and Solutions. New York:
McGraw-Hill, 1994.
Fabozzi, F. J. Fixed-Income Mathematics: Analytical and Statistical Techniques.
New York: McGraw-Hill, 1996.
Fabozzi, F.J. Duration, Convexity, and Other Bond Risk Measures, Frank J.
Fabozzi Associates, 1999.
Grinblatt, M., and F.A. Longstaff. "Financial Innovation and the Role of
Derivatives Securities: An Empirical Analysis of the Treasury Strips
Program," Journal of Finance, 55, 3 (2000): 1415-1436.
Jorion, P. Big Bets Gone Bad: Derivatives and Bankruptcy in Orange County.
New York: Academic Press, 1995.
Stigum, M., and F. L. Robinson. Money Markets and Bond Calculations.
Chicago: Irwin, 1996.
QUESTIONS A N D PROBLEMS (Answers at End of Book)
4.1. A bank quotes you an interest rate of 14% per annum with quarterly
compounding. What is the equivalent rate with (a) continuous compounding and (b) annual compounding?
4.2. An investor receives $1,100 in one year in return for an investment of
$1,000 now. Calculate the percentage return per annum with (a) annual
Interest Rate Risk
107
compounding, (b) semiannual compounding, (c) monthly compounding,
and (d) continuous compounding.
4.3. A deposit account pays 12% per annum with continuous compounding,
but interest is actually paid quarterly. How much interest will be paid each
quarter on a $10,000 deposit?
4.4. What rate of interest with continuous compounding is equivalent to 15%
per annum with monthly compounding?
4.5. Suppose that zero interest rates with continuous compounding are as
follows:
Maturity
{years)
Rate
(% per annum)
1
2
3
4
5
2.0
3.0
3.7
4.2
4.5
Calculate forward interest rates for the second, third, fourth, and fifth
years.
4.6. Suppose that zero interest rates with continuous compounding are as
follows:
Maturity
(months)
Rate
(% per annum)
3
6
9
12
15
18
8.0
8.2
8.4
8.5
8.6
8.7
Calculate forward interest rates for the second, third, fourth, fifth, and
sixth quarters.
4.7. The term structure of interest rates is upward sloping. Put the following in
order of magnitude: (a) the five-year zero rate, (b) the yield on a five-year
coupon-bearing bond, and (c) the forward rate corresponding to the
period between 5 and 5.25 years in the future. What is the answer to this
question when the term structure of interest rates is downward sloping?
4.8. The six-month and one-year zero rates are both 10% per annum. For a
bond that has a life of 18 months and pays a coupon of 8% per annum
108
Chapter 4
(with semiannual payments and one having just been made), the yield is
10.4% per annum. What is the bond's price? What is the 18-month zero
rate? All rates are quoted with semiannual compounding.
4.9. Suppose that 6-month, 12-month, 18-month, 24-month, and 30-month
zero rates are 4%, 4.2%, 4.4%, 4.6%, and 4.8% per annum, respectively,
with continuous compounding. Estimate the cash price of a bond with a
face value of 100 that will mature in 30 months and pays a coupon of 4%
per annum semiannually.
4.10. A three-year bond provides a coupon of 8% semiannually and has a cash
price of 104. What is the bond's yield?
4.11. Why are US Treasury rates significantly lower than other rates that are
close to risk free?
4.12. What does duration tell you about the sensitivity of a bond portfolio to
interest rates. What are the limitations of the duration measure?
4.13. A five-year bond with a yield of 11% (continuously compounded) pays an
8% coupon at the end of each year. (a) What is the bond's price? (b) What
is the bond's duration? (c) Use the duration to calculate the effect on the
bond's price of a 0.2% decrease in its yield. (d) Recalculate the bond's
price on the basis of a 10.8% per annum yield and verify that the result is
in agreement with your answer to (c).
4.14. Repeat Problem 4.13 on the assumption that the yield is compounded
annually. Use modified durations.
4.15. A six-year bond with a continuously compounded yield of 4% provides a
5% coupon at the end of each year. Use duration and convexity to
estimate the effect of a 1% increase in the yield on the price of the bond.
How accurate is the estimate?
4.16. Explain three ways in which a vector of deltas can be calculated to manage
nonparallel yield curve shifts.
4.17. Estimate the delta of the portfolio in Table 4.8 with respect to the first two
factors in Table 4.9.
ASSIGNMENT QUESTIONS
4.18. An interest rate is quoted as 5% per annum with semiannual compounding. What is the equivalent rate with (a) annual compounding, (b) monthly
compounding, and (c) continuous compounding.
4.19. Portfolio A consists of a 1-year zero-coupon bond with a face value of
$2,000 and a 10-year zero-coupon bond with a face value of $6,000.
Portfolio B consists of a 5.95-year zero-coupon bond with a face value
of $5,000. The current yield on all bonds is 10% per annum (continuously
Interest Rate Risk
109
compounded). (a) Show that both portfolios have the same duration. (b)
Show that the percentage changes in the values of the two portfolios for a
0.1% per annum increase in yields are the same. (c) What are the
percentage changes in the values of the two portfolios for a 5% per annum
increase in yields?
4.20. What are the convexities of the portfolios in Problem 4.19? To what extent
does (a) duration and (b) convexity explain the difference between the
percentage changes calculated in part (c) of Problem 4.19?
4.21. When the partial durations are as in Table 4.7 estimate the effect of a shift
in the yield curve where the 10-year rate stays the same, the 1-year rate
moves up by 9e and the movements in intermediate rates are calculated by
interpolation between 9e and 0. How could your answer be calculated
from the results for a rotation presented in Section 4.8?
4.22. Suppose that the change in a portfolio value for a 1-basis-point shift in the
3-month, 6-month, 1-year, 2-year, 3-year, 4-year, and 5-year rates are (in
$million) +5, —3, — 1, +2, +5, +7, and +8, respectively. Estimate the delta
of the portfolio with respect to the first three factors in Table 4.9. Quantify
the relative importance of the three factors for this portfolio.
C H A P T E R
Volatility
Hedging schemes such as those described in the last two chapters eliminate much of the risks from trading activities. This is because traders are
required to ensure that Greek letters such as delta, gamma, and vega are
within certain limits. But trading portfolios are not totally free of risk. At
any given time, a financial institution still has a residual exposure to
changes in hundreds or even thousands of market variables such as
interest rates, exchange rates, equity prices, and commodity prices. The
volatility of a market variable measures uncertainty about the future
value of the variable. It is important for risk managers to monitor the
volatilities of market variables in order to assess potential losses. This
chapter describes the procedures they use to carry out the monitoring.
We begin by defining volatility and then explain how volatility can be
implied from option prices or estimated from historical data. The common assumption that percentage returns from market variables are
normally distributed is examined and we present the power law as an
alternative. After that we move on to consider models with imposing
names such as exponentially weighted moving average (EWMA), autoregressive conditional heteroscedasticity (ARCH), and generalized autoregressive conditional heteroscedasticity (GARCH). The distinctive
feature of these models is that they recognize that volatility is not
constant: during some periods it may be relatively low, whereas during
others it may be relatively high. The models attempt to keep track of the
variations in the volatility through time.
112
Chapter 5
5.1 DEFINITION OF VOLATILITY
The volatility of a variable is defined as the standard deviation of the
return provided by the variable per unit of time when the return is
expressed using continuous compounding. When volatility is used for
option pricing, the unit of time is usually one year, so that volatility is the
standard deviation of the continuously compounded return per year.
However, when volatility is used for risk management, the unit of time
is usually one day, so that volatility is the standard deviation of the
continuously compounded return per day.
In general,
is equal to the standard deviation of
, where
is the value of the market variable at time T and
is its value today.
The expression
equals the total return (not the return per unit
time) earned in time T expressed with continuous compounding. If is
per day, T is measured in days; if is per year, T is measured in years.
When T is small, the continuously compounded return of a market
variable is close to the percentage change. It follows that, for small T,
is approximately equal to the standard deviation of the percentage
change in the market variable in time T. Suppose, for example, that a
stock price is $50 and its volatility is 30% per year. The standard
deviation of the percentage change in the stock price in one week is
approximately
30
= 4.16%
A one-standard-deviation move in the stock price in one week is therefore
50 0.0416, or $2.08.
When the time horizons considered are short, our uncertainty about a
future stock price, as measured by its standard deviation, increases (at least
approximately) with the square root of how far ahead we are looking. For
example, the standard deviation of the stock price in four weeks is
approximately twice the standard deviation in one week. This corresponds
to the adage "uncertainty increases with the square root of time".
Variance Rate
Risk managers often focus on the variance rate rather than the volatilityThe variance rate is defined as the square of the volatility. The variance
rate per year is the variance of the continuously compounded return in
one year; the variance rate per day is the variance of the continuously
compounded return in one day. Whereas the standard deviation of the
return in time T increases with the square root of time, the variance of
Volatility
113
Business Snapshot 5.1
What Causes Volatility?
It is natural to assume that the volatility of a stock price is caused by new
information reaching the market. This information causes people to revise their
opinions on the value of the stock. The price of the stock changes and volatility
results. However, this view of what causes volatility is not supported by
research. With several years of daily stock price data, researchers can calculate:
1. The variance of stock price returns between the close of trading on one
day and the close of trading on the next day when there are no
intervening nontrading days
2. The variance of the stock price returns between the close of trading on
Friday and the close of trading on Monday
The second variance is the variance of returns over a three-day period. The
first is a variance over a one-day period. We might reasonably expect the
second variance to be three times as great as the first variance. Fama (1965),
French (1980), and French and Roll (1986) show that this is not the case.
These three research studies estimate the second variance to be 22%, 19%, and
10.7% higher than the first variance, respectively.
At this stage you might be tempted to argue that these results are explained
by more news reaching the market when the market is open for trading. But
research by Roll (1984) does not support this explanation. Roll looked at the
prices of orange juice futures. By far the most important news for orange juice
futures prices is news about the weather and news about the weather is equally
likely to arrive at any time. When Roll did a similar analysis to that just
described for stocks, he found that the second (Friday-to-Monday) variance is
only 1.54 times the first variance.
The only reasonable conclusion from all this is that volatility is to a large
extent caused by trading itself. (Traders usually have no difficulty accepting
this conclusion!)
this return increases linearly with time. If we wanted to be pedantic, we
could say that it is correct to talk about the variance rate per day but that
volatility is "per square root of day".
Trading Days vs. Calendar Days
When volatilities are calculated and used, an issue that crops up is
whether time should be measured in calendar days or trading days. As
shown in Business Snapshot 5.1, research shows that volatility is much
higher when the exchange is open for trading than when it is closed. As a
result, when estimating volatility from historical data, analysts tend to
114
Chapter 5
ignore days when the exchange is closed. The usual assumption is that
there are 252 days per year.
Define
as the volatility per year of a certain asset, while
is the
equivalent volatility per day of the asset. The standard deviation of the
continuously compounded return on the asset in one year is either
or
. It follows that
with the result that the daily volatility is about 6% of annual volatility.
5.2 IMPLIED VOLATILITIES
As shown in Appendix C at the end of this book, the one parameter in
option pricing formulas that cannot be directly observed is the volatility
of the asset price. This allows traders to imply a volatility from option
prices.
To illustrate how implied volatilities are calculated, suppose that the
market price of a three-month European call option on a non-dividendpaying stock is $1.875 when the stock price is $21, the strike price is $20
and the risk-free rate is 10%. The implied volatility is the value of
volatility that, when substituted into the Black-Scholes option pricing
formula, gives an option price of $1.875. Unfortunately, it is not possible
to invert the Black-Scholes formula so that volatility is expressed as a
function of the option price and other variables. However, an iterative
search procedure can be used to find the implied volatility . For example,
we can start by trying =0.20. This gives a value for the option price
equal to $1.76, which is less than the market price of $1.875. Since the
option price is an increasing function of , a higher value of is required.
We can next try = 0 . 3 . This gives a value for the option price equal to
$2.10, which is too high and means that must lie between 0.20 and 0.30.
Next, we try a value of 0.25 for . This also proves to be too high,
showing that the implied volatility lies between 0.20 and 0.25. Proceeding
in this way, we can halve the range of values for at each iteration and
calculate its correct value to any required accuracy.1 In this example, the
1
This method is presented for illustration. Other more powerful search procedures are
used to calculate implied volatilities in practice.
Volatility
115
implied volatility is 0.235, or 23.5% per annum. A similar procedure can
be used in conjunction with binomial trees to find implied volatilities for
American options.
Implied volatilities are used extensively by traders, as we will explain in
Chapter 15. However, risk management is largely based on historical
volatilities. The rest of this chapter will be concerned with developing
procedures for using historical data to monitor volatility.
5.3 ESTIMATING VOLATILITY FROM HISTORICAL
DATA
When the volatility of a variable is estimated using historical data, it is
usually observed at fixed intervals of time (e.g., every day, week, or month).
Define:
n + 1: Number of observations
Si: Value of variable at end of ith interval, where i = 0, 1 , . . . , n
: Length of time interval
and let
for i = 1,2, ...,n.
The usual estimate s of the standard deviation of the ui is given by
or
where is the mean of the ui.
As explained in Section 5.1, the standard deviation of the ui is
where is the volatility of the variable. The variable s is, therefore, an
estimate of
. It follows that itself can be estimated as , where
The standard error of this estimate can be shown to be approximately
116
Chapter 5
. If T is measured in years, the volatility calculated is a volatility per
year; if is measured in days, the volatility that is calculated is a daily
volatility.
Example 5.1
Table 5.1 shows a possible sequence of stock prices during 21 consecutive
trading days. In this case,
and
and the estimate of the standard deviation of the daily return is
or 1.216%. To calculate a daily volatility, we set = 1 and obtain a volatility
of 1.216%. To calculate a volatility per year, we set = 1/252 and the data
Table 5.1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
20.00
20.10
19.90
20.00
20.50
20.25
20.90
20.90
20.90
20.75
20.75
21.00
21.10
20.90
20.90
21.25
21.40
21.40
21.25
21.75
22.00
Computation of volatility.
1.00500
0.99005
1.00503
1.02500
0.98780
1.03210
1.00000
1.00000
0.99282
1.00000
1.01205
1.00476
0.99052
1.00000
1.01675
1.00706
1.00000
0.99299
1.02353
1.01149
0.00499
-0.01000
0.00501
0.02469
-0.01227
0.03159
0.00000
0.00000
-0.00720
0.00000
0.01198
0.00475
-0.00952
0.00000
0.01661
0.00703
0.00000
-0.00703
0.02326
0.01143
117
Volatility
give an estimate for the volatility per annum of 0.01216
19.3%. The standard error of the daily volatility estimate is
=0.193, or
or 0.19% per day. The standard error of the volatility per year is
or 3.1% per annum.
5.4 ARE DAILY PERCENTAGE CHANGES IN
FINANCIAL VARIABLES NORMAL?
The Black-Scholes model and its extensions (see Appendix C) make the
assumption that asset prices change continuously and have constant
volatility. This means that the return in a short period of time
always
has a normal distribution with a standard deviation of
. Suppose
that the volatility of an exchange rate is estimated as 12% per year. This
corresponds to 12/
, or 0.756%, per day. Assuming normally distributed returns, we see from the tables at the end of this book that the
probability of the value of the foreign currency changing by more than
one standard deviation (i.e., by more than 0.756%) in one day is 31.73%;
the probability that the exchange rate will change by more than two
standard deviations (i.e., by more than 1.512%) is 4.55%; the probability
that it will change by more than three standard deviations (i.e., by more
than 2.268%) is 0.27%; and so on. 2
In practice, exchange rates, as well as most other market variables, tend
to have heavier tails than the normal distribution. Table 5.2 illustrates this
by examining the daily movements in 12 different exchange rates over a
ten-year period. 3 The first step in the production of this table is to
calculate the standard deviation of daily percentage changes in each
exchange rate. The next stage is to note how often the actual percentage
changes exceeded one standard deviation, two standard deviations, and
so on. These numbers are then compared with the corresponding numbers for the normal distribution.
2
We are making a small approximation here that the one-day continuously compounded
return is the same as the one-week return with daily compounding.
. This table is taken from J. C. Hull and A. White, "Value at Risk When Daily Changes
in Market Variables Are Not Normally Distributed." Journal of Derivatives, 5, No. 3
(Spring 1998): 9-19.
Chapter 5
118
Business Snapshot 5.2
Making Money from Foreign Currency Options
Suppose that most market participants think that exchange rates are lognormally distributed. They will be comfortable using the same volatility to
value all options on a particular exchange rate. You have just done the
analysis in Table 5.2 and know that the lognormal assumption is not a good
one for exchange rates. What should you do?
The answer is that you should buy deep-out-of-the-money call and put
options on a variety of different currencies—and wait. These options will be
relatively inexpensive and more of them will close in the money than the
lognormal model predicts. The present value of your payoffs will on average
be much greater than the cost of the options.
In the mid-1980s a few traders knew about the heavy tails of foreign
exchange probability distributions. Everyone else thought that the lognormal
assumption of Black-Scholes was reasonable. The few traders who were well
informed followed the strategy we have described—and made lots of money.
By the late 1980s everyone realized that out-of-the money options should have
a higher implied volatility than at-the-money options and the trading opportunities disappeared.
Daily percentage changes exceed three standard deviations on 1.34% of
the days. The normal model for returns predicts that this should happen
on only 0.27% of days. Daily percentage changes exceed four, five, and
six standard deviations on 0.29%, 0.08%, and 0.03% of days, respectively. The normal model predicts that we should hardly ever observe this
happening. The table, therefore, provides evidence to support the existence of heavy tails. Business Snapshot 5.2 shows how you could have
made money if you had done the analysis in Table 5.2 in 1985!
Table 5.2 Percentage of days when absolute size of daily
exchange rate moves is greater than 1, 2 , . . . , 6 standard deviations.
(SD = standard deviation of daily percentage change.)
>1 SD
>2 SD
>3 SD
>4 SD
>5 SD
>6 SD
Real world
(%)
Normal model
(%)
25.04
5.27
1.34
0.29
0.08
0.03
31.73
4.55
0.27
0.01
0.00
0.00
Volatility
119
Figure 5.1 Comparison of normal distribution with a
heavy-tailed distribution. The two distributions have the
same mean and standard deviation.
Figure 5.1 compares a typical heavy-tailed distribution (such as the one
for foreign exchange) with a normal distribution that has the same mean
and standard deviation.4 The heavy-tailed distribution is more peaked
than the normal distribution. In Figure 5.1, we can distinguish three parts
of the distribution: the middle, the tails, the intermediate parts (between
the middle and the tails). When we move from the normal distribution to
the heavy-tailed distribution, probability mass shifts from the intermediate
parts of the distribution to both the tails and the middle. If we are
considering the percentage change in a market variable, the heavy-tailed
distribution has the property that small and large changes in the variable
are more likely than they would be if a normal distribution were assumed.
Intermediate changes are less likely.
An Alternative to Normal Distributions: The Power Law
The power law asserts that, for many variables that are encountered in
practice, it is approximately true that the value v of the variable has the
Property that, when x is large,
(5.1)
where K and
are constants. This equation has been found to be
4
Kurtosis measures the size of a distribution's tails. A leptokurtic distribution has heavier
tails than the normal distribution. A Platykurtic distribution has less heavy tails than the
normal distribution. A distribution with tails of the same size as the normal distribution is
termed mesokurtic.
120
Chapter 5
approximately true for variables as diverse as the income of individuals,
the size of cities, and the number of visits to a website. Suppose that = 3
and we observe that the probability that
is 0.05. In this case
K = 50 and we can estimate that the probability that
is 0.00625;
the probability that
is 0.0019; and so on.
Equation (5.1) implies that
We can therefore do a quick test of whether it holds by plotting
against
. We do this for our exchange rate data in
Figure 5.2. The logarithm of the probability of the exchange rate increasing
by more than x standard deviations is approximately linearly dependent on
for x > 3 showing that the power law holds. The parameter is about
5.5. When producing Figure 5.2, we assume that the distribution of
exchange rate changes in Table 5.2 is symmetrical, so that the probability
of a change greater than one standard deviation is 0.5 25.04 = 12.52%,
greater than two standard deviations is 0.5 5.27 = 2.635%, and so on.
We will examine the power law more formally and explain better
Figure 5.2
Log-log plot for exchange rate increases: x is number of
standard deviations; v is the exchange rate increase.
Volatility
121
procedures for estimating the parameters when we consider extreme value
theory in Chapter 9. We will also explain how it can be used in the
assessment of operational risk in Chapter 14.
5.5 MONITORING DAILY VOLATILITY
Risk managers cannot assume that asset prices are well behaved with a
constant volatility. It is important for them to monitor volatility daily.
Define
as the volatility of a market variable on day n, as estimated at
the end of day n — 1. The variance rate is . Suppose that the value of
the market variable at the end of day i is Si. As in Section 5.3, we define
the variable
as the continuously compounded return during day i
(between the end of day i — 1 and the end of day i), so that
One approach to estimating is to set it equal to the historical volatility as
calculated in Section 5.3. When m days of observations on the are used,
this approach gives
(5.2)
where
is the mean of the
:
For risk management purposes, the formula in equation (5.2) is usually
changed in a number of ways:
1.
is defined as the percentage change in the market variable
between the end of day i — 1 and the end of day i so that
(5.3)
This makes very little difference to the values of ui that are computed.
2. is assumed to be zero. The justification for this is that the expected
change in a variable in one day is very small when compared with
the standard deviation of changes.
3. m — 1 is replaced by m. This moves us from an unbiased estimate of
volatility to a maximum-likelihood estimate (see Section 5.9).
122
Chapter 5
These three changes allow the formula for the variance rate to be
simplified to
(5.4)
where
is given by equation (5.3).
Weighting Schemes
Equation (5.4) gives equal weight to all
. Our objective is to estimate the current level of volatility,
. It therefore makes
sense to give more weight to recent data. A model that does this is
(5.5)
The variable
is the amount of weight given to the observation days
ago. The 's are positive. If we choose them so that
when
less weight is given to older observations. The weights must sum to unity,
so that
An extension of the idea in equation (5.5) is to assume that there is a
long-run average variance rate and that this should be given some weight.
This leads to the model that takes the form
(5.6)
where
is the long-run variance rate and is the weight assigned to
Since the weights must sum to unity, we have
This is known as an ARCH(m) model. It was first suggested by Engle.
The estimate of the variance is based on a long-run average variance
and m observations. The older an observation, the less weight it is given.
5
See R. Engle, "Autoregressive Conditional Heteroscedasticity with Estimates of the
Variance of UK Inflation," Econometrica, 50 (1982), 987-1008. Robert Engle won the
Nobel prize for economics in 2003 for his work on ARCH models.
123
Volatility
Defining
, we can write the model in equation (5.6) as
In the next two sections we discuss two important approaches to monitoring volatility using the ideas in equations (5.5) and (5.6).
5.6 THE EXPONENTIALLY WEIGHTED MOVING
AVERAGE MODEL
The exponentially weighted moving average (EWMA) model is a particular case of the model in equation (5.5) where the weights
decrease
exponentially as we move back through time. Specifically,
,
where is a constant between 0 and 1.
It turns out that this weighting scheme leads to a particularly simple
formula for updating volatility estimates. The formula is
The estimate
of the volatility for day n (made at the end of day n — 1)
is calculated from
(the estimate of the volatility for day n — 1 that
was made at the end of day n — 2) and
(the most recent daily
percentage change).
To understand why equation (5.8) corresponds to weights that decrease
exponentially, we substitute for
to get
or
Substituting in a similar way for
gives
Continuing in this way, we see that
For a large m, the term
is sufficiently small to be ignored, so that
124
Chapter 5
equation (5.8) is the same as equation (5.5) with
. The
weights for the u's decline at rate as we move back through time. Each
weight is times the previous weight.
Example 5.2
Suppose that is 0.90, the volatility estimated for a market variable for day
n — 1 is 1 % per day, and during day n — 1 the market variable increased by
2%. This means that
= 0.012 = 0.0001 and
= 0.022 = 0.0004. Equation (5.8) gives
= 0.9 0.0001+0.1 0.0004 = 0.00013
The estimate of the volatility for day n is therefore
, or 1.14%, per
day. Note that the expected value of
is
, or 0.0001. In this example,
the realized value of
is greater than the expected value, and as a result
our volatility estimate increases. If the realized value of
had been less
than its expected value, our estimate of the volatility would have decreased.
The EWMA approach has the attractive feature that relatively little data
need to be stored. At any given time, we need to remember only the
current estimate of the variance rate and the most recent observation on
the value of the market variable. When we get a new observation on the
value of the market variable, we calculate a new daily percentage change
and use equation (5.8) to update our estimate of the variance rate. The
old estimate of the variance rate and the old value of the market variable
can then be discarded.
The EWMA approach is designed to track changes in the volatility.
Suppose there is a big move in the market variable on day n — 1, so that
is large. From equation (5.8), this causes our estimate of the current
volatility to move upward. The value of governs how responsive the
estimate of the daily volatility is to the most recent daily percentage change.
A low value of leads to a great deal of weight being given to the
when
is calculated. In this case, the estimates produced for the volatility on
successive days are themselves highly volatile. A high value of (i.e., a
value close to 1.0) produces estimates of the daily volatility that respond
relatively slowly to new information given by the daily percentage change.
The RiskMetrics database, which was originally created by J. P. Morgan
and made publicly available in 1994, uses the EWMA model with = 0.94
for updating daily volatility estimates. The company found that, across a
range of different market variables, this value of gives forecasts of the
variance rate that come closest to the realized variance rate.6 The realized
6
See J. P. Morgan, RiskMetrics Monitor, Fourth Quarter, 1995. We will explain an
alternative (maximum-likelihood) approach to estimating parameters later in the chapter.
Volatility
125
variance rate on a particular day was calculated as an equally weighted
average of the
on the subsequent 25 days (see Problem 5.20).
5.7 THE GARCH(1,1) MODEL
We now move on to discuss what is known as the GARCH(1,1) model,
proposed by Bollerslev in 1986.7 The difference between the GARCH(1,1)
model and the EWMA model is analogous to the difference between
equation (5.5) and equation (5.6). In GARCH(1,1),
is calculated from
a long-run average variance rate,
, as well as from
and
. The
equation for GARCH(1,1) is
(5.9)
where is the weight assigned to , is the weight assigned to
, and
is the weight assigned to
. Since the weights must sum to one, we
have
The EWMA model is a particular case of GARCH(1,1) where
,
, and
The "(1,1)" in GARCH(1,1) indicates that
is based on the most
recent observation of and the most recent estimate of the variance rate.
The more general GARCH(p,q) model calculates
from the most recent
observations on and the most recent estimates of the variance rate.8
GARCH(1, 1) is by far the most popular of the GARCH models.
Setting
, we can also write the GARCH(1,1) model as
(5.10)
This is the form of the model that is usually used for the purposes of
7
See T. Bollerslev. "Generalized Autoregressive Conditional Heteroscedasticity,"
Journal of Econometrics, 31 (1986), 307-327.
8
Other GARCH models have been proposed that incorporate asymmetric news. These
models are designed so that
depends on the sign of
. Arguably, these models are
more appropriate than GARCH(1,1) for equities. This is because the volatility of an
equity's price tends to be inversely related to the price, so that a negative
should
have a bigger effect on
than the same positive
. For a discussion of models for
handling asymmetric news, see D. Nelson, "Conditional Heteroscedasticity and Asset
Returns; A New Approach," Econometrica, 59 (1990), 347-370 and R. F. Engle and
V. Ng, "Measuring and Testing the Impact of News on Volatility," Journal of Finance, 48
(1993), 1749-1778.
126
Chapter 5
estimating the parameters. Once
and have been estimated, we can
calculate as
The long-term variance VL can then be calculated as
For a stable GARCH(1,1) process, we require
< 1.
Otherwise the weight applied to the long-term variance is negative.
Example 5.3
Suppose that a GARCH(1,1) model is estimated from daily data as
This corresponds to =0.13,
=0.86, and
=0.000002. Since
=
1it follows that = 0.01, and, since
we have VL = 0.0002.
In other words, the long-run average variance per day implied by the model is
0.0002. This corresponds to a volatility of
= 0.014, or 1.4%, per day.
Suppose that the estimate of the volatility on day n - 1 is 1.6% per day, so
that
= 0.0162 = 0.000256, and that on day n - 1 the market variable
decreased by 1%, so that
= 0.012 = 0.0001. Then
= 0.000002 + 0.13 x 0.0001 + 0.86 x 0.000256 = 0.00023516
The new estimate of the volatility is therefore
per day.
= 0.0153, or 1.53%,
The Weights
Substituting for
in equation (5.10), we obtain
or
Substituting for
we get
Continuing in this way, we see that the weight applied to
is
The weights decline exponentially at rate
The parameter
can be
interpreted as a "decay rate". It is similar to in the EWMA model. It
defines the relative importance of the u's in determining the current
variance rate. For example, if = 0.9, then
is only 90% as important as
is 8 1 % as important as
and so on. The
GARCH(1,1) model is the same as the EWMA model except that, in
addition to assigning weights that decline exponentially to past
it also
assigns some weight to the long-run average volatility.
Volatility
127
5.8 CHOOSING BETWEEN THE MODELS
In practice, variance rates do tend to be pulled back to a long-run average
level. This is known as mean reversion. The GARCH(1,1) model incorporates mean reversion whereas the EWMA model does not. GARCH(1,1) is
therefore theoretically more appealing than the EWMA model.
In the next section, we shall discuss how best-fit parameters
and
in GARCH(1,1) can be estimated. When the parameter
is zero, the
GARCH(1,1) reduces to EWMA. In circumstances where the best-fit
value of turns out to be negative, the GARCH(1,1) model is not stable
and it makes sense to switch to the EWMA model.
5.9 MAXIMUM-LIKELIHOOD METHODS
It is now appropriate to discuss how the parameters in the models we have
been considering are estimated from historical data. The approach used is
known as the maximum-likelihood method. It involves choosing values for
the parameters that maximize the chance (or likelihood) of the data
occurring.
To illustrate the method, we start with a very simple example. Suppose
that we sample ten stocks at random on a certain day and find that the
price of one of them declined on that day and the prices of the other nine
either remained the same or increased. What is our best estimate of the
probability of a price decline? The natural answer is 0.1. Let us see if this
is the result given by the maximum-likelihood method.
Suppose that the probability of a price decline is p. The probability
that one particular stock declines in price and the other nine do not is
p(l - p)9. (There is a probability p that it will decline and 1 — p that each
of the other nine will not.) Using the maximum-likelihood approach, the
best estimate of p is the one that maximizes p(1 — p)9. Differentiating this
expression with respect to p and setting the result equal to zero, we find
that p = 0.1 maximizes the expression. This shows that the maximumlikelihood estimate of p is 0.1, as expected.
Estimating a Constant Variance
As our next example of maximum-likelihood methods, we consider the
problem of estimating the variance of a variable X from m observations
on X when the underlying distribution is normal with mean zero. We
assume that the observations are u1, u2, ..., um and that the mean of the
128
Chapter 5
underlying distribution is zero. Denote the variance by The likelihood
of
being observed is the probability density function for X when
X=
This is
The likelihood of m observations occurring in the order in which they are
observed is
Using the maximum-likelihood method, the best estimate of is the value
that maximizes this expression.
Maximizing an expression is equivalent to maximizing the logarithm of
the expression. Taking logarithms of the expression in equation (5.11)
and ignoring constant multiplicative factors, it can be seen that we wish to
maximize
or
Differentiating this expression with respect to and setting the resultant
equation to zero, we see that the maximum-likelihood estimator of is9
Estimating GARCH(1,1) Parameters
We now consider how the maximum-likelihood method can be used to
estimate the parameters when GARCH(1,1) or some other volatility
updating scheme is used. Define
as the estimated variance for
day i. We assume that the probability distribution of conditional on the
variance is normal. A similar analysis to the one just given shows the best
The unbiased estimator has m replaced by m — 1.
129
Volatility
parameters are the ones that maximize
Taking logarithms, we see that this is equivalent to maximizing
This is the same as the expression in equation (5.12), except that is
replaced by
We search iteratively to find the parameters in the model
that maximize the expression in equation (5.13).
The spreadsheet in Table 5.3 indicates how the calculations could be
organized for the GARCH(1,1) model. The table analyzes data on the
Japanese yen exchange rate between January 6, 1988, and August 15,
1997. The numbers in the table are based on trial estimates of the three
GARCH(1,1) parameters:
and
The first column in the table
records the date. The second counts the days. The third shows the
exchange rate
at the end of day i. The fourth shows the proportional
change in the exchange rate between the end of day i - 1 and the end of
Table 5.3
Estimation of parameters in GARCH(1,1) model.
Date
Day i
Si
ui
06-Jan-88
07-Jan-88
08-Jan-88
ll-Jan-88
12-Jan-88
13-Jan-88
1
2
3
4
5
6
0.007728
0.007779
0.007746
0.007816
0.007837
0.007924
0.006599
-0.004242
0.009037
0.002687
0.011101
0.00004355
0.00004198
0.00004455
0.00004220
9.6283
8.1329
9.8568
7.1529
13-Aug-97
14-Aug-97
15-Aug-97
2421
2422
2423
0.008643
0.008493
0.008495
0.003374
-0.017309
0.000144
0.00007626
0.00007092
0.00008417
9.3321
5.3294
9.3824
22,063.5763
Trial estimates of GARCH parameters
0.00000176
0.0626
0.8976
130
Chapter 5
day i. This is
The fifth column shows the estimate of
the variance rate
for day i made at the end of day i — 1. On day
three, we start things off by setting the variance equal to
On subsequent days equation (5.10) is used. The sixth column tabulates the
likelihood measure,
The values in the fifth and sixth
columns are based on the current trial estimates of
and
We are
interested in choosing
and to maximize the sum of the numbers in
the sixth column. This involves an iterative search procedure.10
In our example, the optimal values of the parameters turn out to be
= 0.00000176,
=0.0626,
=0.8976
and the maximum value of the function in equation (5.13) is 22,063.5763.
The numbers shown in Table 5.3 were calculated on the final iteration of
the search for the optimal
and
The long-term variance rate, VL, in our example is
The long-term volatility is
or 0.665%, per day.
Figure 5.3 shows the way in which the GARCH(1,1) volatility for the
Japanese yen changed over the ten-year period covered by the data. Most
of the time, the volatility was between 0.4% and 0.8% per day, but
volatilities over 1% were experienced during some periods.
An alternative and more robust approach to estimating parameters in
GARCH(1,1) is known as variance targeting.11 This involves setting the
long-run average variance rate, VL, equal to the sample variance calculated from the data (or to some other value that is believed to be
reasonable.) The value of
then equals
and only two
parameters have to be estimated. For the data in Table 5.3 the sample
variance is 0.00004341, which gives a daily volatility of 0.659%. Setting
VL equal to the sample variance, we find that the values of
that
maximize the objective function in equation (5.13) are 0.0607 and 0.8990,
respectively. The value of the objective function is 22,063.5274, only
10
As discussed later, a general purpose algorithm such as Solver in Microsoft's Excel
can be used. Alternatively, a special purpose algorithm, such as Levenberg-Marquardt,
can be used. See, for example, W. H. Press, B.P. Flannery, S.A. Teukolsky, and
W. T. Vetterling. Numerical Recipes in C: The Art of Scientific Computing, Cambridge
University Press, 1988.
11
See R. Engle and J. Mezrich, "GARCH for Groups," Risk, August 1996, 36-40.
131
Volatility
Jan-88
Jan-89
Jan-90
Jan-91
Jan-92
Jan-93
Jan-94
Jan-95
Jan-96
Jan-97
Jan-98
Figure 5.3 Daily volatility of the yen/USD exchange rate, 1988-97.
marginally below the value of 22,063.5763 obtained using the earlier
procedure.
When the EWMA model is used, the estimation procedure is relatively
simple. We set = 0 , = 1 — , and
and only one parameter has
to be estimated. In the data in Table 5.3, the value of that maximizes the
objective function in equation (5.13) is 0.9686 and the value of the
objective function is 21,995.8377.
Both GARCH(1,1) and the EWMA method can be implemented by
using the Solver routine in Excel to search for the values of the parameters that maximize the likelihood function. The routine works well
provided that we structure our spreadsheet so that the parameters we are
[searching for have roughly equal values. For example, in GARCH(1,1)
we could let cells Al, A2, and A3 contain x 105, and 0.1 We could
then set Bl = Al/100,000, B2 = A2, and B3 = 10 * A3. We could use
Bl, B2, and B3 to calculate the likelihood function and then ask Solver to
(calculate the values of Al, A2, and A3 that maximize the likelihood
function.
How Good Is the Model?
The assumption underlying a GARCH model is that volatility changes
with the passage of time. During some periods volatility is relatively high;
[during others it is relatively low. To put this another way, when is high,
132
Chapter 5
Table 5.4 Autocorrelations before and after the
use of a GARCH model.
Time lag
Autocorrelation
for
Autocorrelation
for
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.072
0.041
0.057
0.107
0.075
0.066
0.019
0.085
0.054
0.030
0.038
0.038
0.057
0.040
0.007
0.004
-0.005
0.008
0.003
0.016
0.008
-0.033
0.012
0.010
-0.023
-0.004
-0.021
-0.001
0.002
-0.028
'„
there is a tendency for
to be high; when
is low, there is a
tendency for
to be low. We can test how true this is by
examining the autocorrelation structure of the
Let us assume that the
do exhibit autocorrelation. If a GARCH
model is working well, it should remove the autocorrelation. We can test
whether it has done this by considering the autocorrelation structure for
the variables
If these show very little autocorrelation, our model
foi
has succeeded in explaining autocorrelations in the
Table 5.4 shows results for the yen/USD exchange rate data referred to
earlier. The first column shows the lags considered when the autocorrelation is calculated. The second column shows autocorrelations for
the
third column shows autocorrelations for
The table shows that
the autocorrelations are positive for
for all lags between 1 and 15. In
the case of
some of the autocorrelations are positive and some are
negative. They tend to be smaller in magnitude than the autocorrelations
for
The GARCH model appears to have done a good job in explaining the
12
For a series
between
and
the autocorrelation with a lag of k is the coefficient of correlation
133
Volatility
data. For a more scientific test, we can use what is known as the LjungBox statistic.13 If a certain series has m observations the Ljung-Box
statistic is
where
is the autocorrelation for a lag of k, K is the number of lags
considered, and
For K = 15, zero autocorrelation can be rejected with 95% confidence
when the Ljung-Box statistic is greater than 25.
From Table 5.4, the Ljung-Box statistic for the
series is about 123.
This is strong evidence of autocorrelation. For the
series the LjungBox statistic is 8.2, suggesting that the autocorrelation has been largely
removed by the GARCH model.
5.10 USING GARCH(1,1) TO FORECAST FUTURE
VOLATILITY
The variance rate estimated at the end of day n — 1 for day n, when
GARCH(l,l) is used, is
so that
On day n + t in the future, we have
The expected value of
Hence,
where E denotes expected value. Using this equation repeatedly yields
134
Chapter 5
Figure 5.4 Expected path for the variance rate when (a) current variance rate is
above long-term variance rate and (b) current variance rate is below long-term
variance rate.
or
This equation forecasts the volatility on day n +t using the information
available at the end of day n — 1. In the EWMA model,
= 1 and
equation (5.14) shows that the expected future variance rate equals the
current variance rate. When
< 1, the final term in the equation
becomes progressively smaller as t increases. Figure 5.4 shows the
expected path followed by the variance rate for situations where the
current variance rate is different from VL. As mentioned earlier, the
variance rate exhibits mean reversion with a reversion level of VL and a
reversion rate of
Our forecast of the future variance rate tends
toward VL as we look further and further ahead. This analysis emphasizes the point that we must have
< 1 for a stable GARCH(1,1)
process. When
> 1, the weight given to the long-term average
variance is negative and the process is "mean fleeing" rather than "mean
reverting".
In the yen/USD exchange rate example considered earlier,
= 0.9602 and VL = 0.00004422. Suppose that our estimate of the
current variance rate per day is 0.00006. (This corresponds to a volatility
of 0.77% per day.) In ten days the expected variance rate is
0.00004422 + 0.960210(0.00006 - 0.00004422) = 0.00005473
Volatility
135
The expected volatility per day is 0.74%, still well above the long-term
volatility of 0.665% per day. However, the expected variance rate in 100
days is
0.00004422 + 0.9602100(0.00006 - 0.00004422) = 0.00004449
and the expected volatility per day is 0.667%, very close the long-term
volatility.
Volatility Term Structures
Suppose it is day n. Define
so that equation (5.14) becomes
Here V(t) is an estimate of the instantaneous variance rate in t days. The
average variance rate per day between today and time T is
The longer the life of the option, the closer this is to VL. Define
as
the volatility per annum that should be used to price a 7-day option
under GARCH(1,1). Assuming 252 days per year,
is 252 times the
average variance rate per day, so that
(5.15)
As we discuss in Chapter 15, the market prices of different options on the
same asset are often used to calculate a volatility term structure. This is the
relationship between the implied volatilities of the options and their
maturities. Equation (5.15) can be used to estimate a volatility term
structure based on the GARCH(1,1) model. The estimated volatility term
structure is not usually the same as the actual volatility term structure.
However, as we will show, it is often used to predict the way that the
actual volatility term structure will respond to volatility changes.
When the current volatility is above the long-term volatility, the
136
Chapter 5
GARCH(1,1) model estimates a downward-sloping volatility term structure. When the current volatility is below the long-term volatility, it
estimates an upward-sloping volatility term structure. In the case of the
yen/USD exchange rate a = ln(l/0.9602) = 0.0406 and VL = 0.00004422.
Suppose that the current variance rate per day, V(0) is estimated as
0.00006 per day. It follows from equation (5.15) that
where T is measured in days. Table 5.5 shows the volatility per year for
different values of T.
Impact of Volatility Changes
Equation (5.15) can be written as
When
changes by
changes
by
Table 5.6 shows the effect of a volatility change on options of varying
maturities for our yen/USD exchange rate example. We assume as before
that V(0) = 0.00006, so that
=12.30%. The table considers a
100-basis-point change in the instantaneous volatility from 12.30% per
year to 13.30% per year. This means that
= 0.01, or 1%.
Many financial institutions use analyses such as this when determining
the exposure of their books to volatility changes. Rather than consider an
across-the-board increase of 1% in implied volatilities when calculating
vega, they relate the size of the volatility increase that is considered to the
maturity of the option. Based on Table 5.6, a 0.84% volatility increase
Table 5.5 Yen/USD volatility term structure predicted from GARCH(1, 1).
Option life (days):
Option volatility (% per annum):
10
30
50
100
500
12.00
11.59
11.33
11.00
10.65
137
Volatility
Table 5.6 Impact of 1% change in the instantaneous volatility predicted from
GARCH(1,1).
Option life (days):
Increase in volatility (%):
10
30
50
100
500
0.84
0.61
0.46
0.27
0.06
would be considered for a 10-day option, a 0.61% increase for a 30-day
option, a 0.46% increase for a 50-day option, and so on.
SUMMARY
In option pricing we define the volatility of a variable as the standard
deviation of its continuously compounded return per year. Volatilities are
either estimated from historical data or implied from option prices. In
risk management the daily volatility of a market variable is defined as the
standard deviation of the percentage daily change in the market variable.
The daily variance rate is the square of the daily volatility. Volatility tends
to be much higher on trading days than on nontrading days. As a result
nontrading days are ignored in volatility calculations. It is tempting to
assume that daily changes in market variables are normally distributed.
In fact, this is far from true. Most market variables have distributions for
percentage daily changes with much heavier tails than the normal distribution. The power law has been found to be a good description of the
tails of many distributions that are encountered in practice, and is often
used for the tails of the distributions of percentage changes in many
market variables.
Most popular option pricing models, such as Black-Scholes, assume
that the volatility of the underlying asset is constant. This assumption is
far from perfect. In practice, the volatility of an asset, like its price, is a
stochastic variable. However, unlike the asset price, it is not directly
observable. This chapter has discussed schemes for attempting to keep
track of the current level of volatility.
We define ui, as the percentage change in a market variable between the
end of day i — 1 and the end of day i. The variance rate of the market
variable (i.e., the square of its volatility) is calculated as a weighted
average of the
The key feature of the schemes that have been discussed
here is that they do not give equal weight to the observations on the
The more recent an observation, the greater the weight assigned to it. In
the EWMA model and the GARCH(1,1) model, the weights assigned to
138
Chapter 5
observations decrease exponentially as the observations become older.
The GARCH(1,1) model differs from the EWMA model in that some
weight is also assigned to the long-run average variance rate. Both the
EWMA and GARCH(1,1) models have structures that enable forecasts
of the future level of variance rate to be produced relatively easily.
Maximum-likelihood methods are usually used to estimate parameters
in GARCH(1,1) and similar models from historical data. These methods
involve using an iterative procedure to determine the parameter values
that maximize the chance or likelihood that the historical data will occur.
Once its parameters have been determined, a model can be judged by how
well it removes autocorrelation from the
FURTHER READING
On the Causes of Volatility
Fama, E. F., "The Behavior of Stock Market Prices," Journal of Business, 38
(January 1965): 34-105.
French, K. R., "Stock Returns and the Weekend Effect," Journal of Financial
Economics, 8 (March 1980): 55-69.
French, K. R, and R. Roll, "Stock Return Variances: The Arrival of
Information and the Reaction of Traders," Journal of Financial Economics,
17 (September 1986): 5-26.
Roll, R., "Orange Juice and Weather," American Economic Review, 74, No. 5
(December 1984): 861-880.
On GARCH
Bollerslev, T., "Generalized Autoregressive Conditional Heteroscedasticity,"
Journal of Econometrics, 31 (1986): 307-327.
Cumby, R., S. Figlewski, and J. Hasbrook, "Forecasting Volatilities and
Correlations with EGARCH Models," Journal of Derivatives, 1, No. 2 (Winter
1993): 51-63.
Engle, R. F., "Autoregressive Conditional Heteroscedasticity with Estimates of
the Variance of UK Inflation," Econometrica, 50 (1982): 987-1008.
Engle, R.F. and J. Mezrich, "Grappling with GARCH," Risk, September 1995:
112-117.
Engle, R. F., and V. Ng, "Measuring and Testing the Impact of News on
Volatility," Journal of Finance, 48 (1993): 1749-1778.
Nelson, D., "Conditional Heteroscedasticity and Asset Returns; A New
Approach," Econometrica, 59 (1990): 347-370.
Volatility
139
Noh, J., R. F. Engle, and A. Kane, "Forecasting Volatility and Option Prices of
the S&P 500 Index," Journal of Derivatives, 2 (1994): 17-30.
QUESTIONS A N D PROBLEMS (Answers at End of Book)
5.1. The volatility of a stock price is 30% per annum. What is the standard
deviation of the percentage price change in one week?
5.2. The volatility of an asset is 25% per annum. What is the standard
deviation of the percentage price change in one trading day. Assuming a
normal distribution, estimate 95% confidence limits for the percentage
price change in one day.
5.3. Why do traders assume 252 rather than 365 days in a year when using
volatilities?
5.4. What is implied volatility? How can it be calculated? In practice, different
options on the same asset have different implied volatilities. What conclusions do you draw from this?
5.5. Suppose that observations on an exchange rate at the end of the last
11 days have been 0.7000, 0.7010, 0.7070, 0.6999, 0.6970, 0.7003, 0.6951,
0.6953, 0.6934, 0.6923, 0.6922. Estimate the daily volatility using both the
approach in Section 5.3 and the simplified approach in equation (5.4).
5.6. The number of visitors to a website follows the power law given in
equation (5.1) with = 2. Suppose that 1 % of sites get 500 or more visitors
per day. What percentage of sites get (a) 1000 and (b) 2000 or more visitors
per day.
5.7. Explain the exponentially weighted moving average (EWMA) model for
estimating volatility from historical data.
5.8. What is the difference between the exponentially weighted moving average
model and the GARCH(1,1) model for updating volatilities?
5.9. The most recent estimate of the daily volatility of an asset is 1.5% and the
price of the asset at the close of trading yesterday was $30.00. The
parameter in the EWMA model is 0.94. Suppose that the price of the
asset at the close of trading today is $30.50. How will this cause the
volatility to be updated by the EWMA model?
5.l0. A company uses an EWMA model for forecasting volatility. It decides to
change the parameter from 0.95 to 0.85. Explain the likely impact on the
forecasts.
5.l1. Assume that S&P 500 at close of trading yesterday was 1,040 and the daily
volatility of the index was estimated as 1% per day at that time. The
parameters in a GARCH(1,1) model are =0.000002, =0.06, and
140
Chapter 5
= 0.92. If the level of the index at close of trading today is 1,060, what is
the new volatility estimate?
5.12. The most recent estimate of the daily volatility of the USD/GBP exchange
rate is 0.6% and the exchange rate at 4 p.m. yesterday was 1.5000. The
parameter in the EWMA model is 0.9. Suppose that the exchange rate at
4 p.m. today proves to be 1.4950. How would the estimate of the daily
volatility be updated?
5.13. A company uses the GARCH(1,1) model for updating volatility. The
three parameters are
and
Describe the impact of making a small
increase in each of the parameters while keeping the others fixed.
5.14. The parameters of a GARCH(1,1) model are estimated as = 0.000004,
= 0.05, and = 0.92. What is the long-run average volatility and what is
the equation describing the way that the variance rate reverts to its longrun average? If the current volatility is 20% per year, what is the expected
volatility in 20 days?
5.15. Suppose that the daily volatility of the FTSE 100 stock index (measured in
GBP) is 1.8% and the daily volatility of the USD/GBP exchange rate is
0.9%. Suppose further that the correlation between the FTSE 100 and the
USD/GBP exchange rate is 0.4. What is the volatility of the FTSE 100
when it is translated to US dollars? Assume that the USD/GBP exchange
rate is expressed as the number of US dollars per pound sterling. (Hint:
When Z = XY, the percentage daily change in Z is approximately equal to
the percentage daily change in X plus the percentage daily change in Y.)
5.16. Suppose that GARCH(1,1) parameters have been estimated as
= 0.000003,
=0.04, and
=0.94. The current daily volatility is
estimated to be 1%. Estimate the daily volatility in 30 days.
5.17. Suppose that
= 0.000002,
estimated to be
used to price a
GARCH(1,1) parameters have been estimated as
=0.04, and =0.94. The current daily volatility is
1.3%. Estimate the volatility per annum that should be
20-day option.
ASSIGNMENT QUESTIONS
5.18. Suppose that observations on a stock price (in US dollars) at the end of
each of 15 consecutive weeks are as follows:
30.2, 32.0, 31.1, 30.1, 30.2, 30.3, 30.6, 33.0, 32.9, 33.0, 33.5, 33.5, 33.7, 33.5, 33.2
Estimate the stock price volatility. What is the standard error of your
estimate?
5.19. Suppose that the price of gold at close of trading yesterday was $300 and
its volatility was estimated as 1.3% per day. The price at the close of
Volatility
141
trading today is $298. Update the volatility estimate using (a) the EWMA
model with = 0.94 and (b) the GARCH(1,1) model with =0.000002,
= 0.04, and = 0.94.
5.20. An Excel spreadsheet containing over 900 days of daily data on a number
of different exchange rates and stock indices can be downloaded from the
author's website: http://www.rotman.utoronto.ca/~hull. Choose one
exchange rate and one stock index. Estimate the value of in the EWMA
model that minimizes the value of
where is the variance forecast made at the end of day i — 1 and is the
variance calculated from data between day i and day i + 25. Use the Solver
tool in Excel. To start the EWMA calculations, set the variance forecast at
the end of the first day equal to the square of the return on that day.
5.21. Suppose that the parameters in a GARCH(1,1) model are = 0 . 0 3 ,
= 0.95, and = 0.000002. (a) What is the long-run average volatility?
(b) If the current volatility is 1.5% per day, what is your estimate of the
volatility in 20, 40, and 60 days? (c) What volatility should be used to price
20-, 40-, and 60-day options? (d) Suppose that there is an event that
increases the current volatility by 0.5% to 2% per day. Estimate the effect
on the volatility in 20, 40, and 60 days. (e) Estimate by how much the
event increases the volatilities used to price 20-, 40-, and 60-day options.
5.22. An Excel spreadsheet containing over 900 days of daily data on a number
of different exchange rates and stock indices can be downloaded from the
author's website: http://www.rotman.utoronto.ca/~hull. Use the data
and maximum-likelihood methods to estimate for the TSE and S&P
indices the best-fit parameters in an EWMA model and a GARCH(1,1)
model for the variance rate.
Correlations
and Copulas
Suppose a company has an exposure to two different market variables. In
the case of each variable it gains $10 million if there is a one-standarddeviation increase and loses $10 million if there is a one-standarddeviation decrease. If changes in the two variables have a high positive
correlation, the company's total exposure is very high; if they have a
correlation of zero, the exposure is less, but still quite large; if they have a
high negative correlation, the exposure is quite low because a loss on one
of the variables is likely to be offset by a gain on the other. This example
shows that it is important for a risk manager to estimate correlations
between the changes in market variables as well as their volatilities when
assessing risk exposures.
This chapter explains how correlations can be monitored in a similar
way to volatilities. It also covers what are known as copulas. These are
tools that provide a way of defining a correlation structure between two
or more variables, regardless of the shapes of their probability distributions. Copulas will prove to be important in a number of future
chapters. For example, a knowledge of copulas enables some of the
formulas underlying the Basel II capital requirements to be understood
(Chapter 7). Copulas are also useful in modeling default correlation for
the purposes of valuing credit derivatives (Chapter 13) and in the
calculation of economic capital (Chapter 16). The final section of this
chapter explains how copulas can be used to model defaults on portfolios of loans.
Chapter 6
144
6.1 DEFINITION OF CORRELATION
The coefficient of correlation,
defined as
between two variables V\ and V2 is
where E(•) denotes expected value and SD(•) denotes standard deviation.
If there is no correlation between the variables, then E{V1V2)=
E(V1)E(V2) and = 0. If V1 = V2, then the numerator and the denominator in the expression for are both equal to the variance of V1. As we
would expect, = 1 in this case.
The Covariance between V1 and V2 is defined as
cov(V1, V2) = E{V1 V2) -
E(V1)E(V2)
(6.2)
so that the correlation can be written
Although it is easier to develop intuition about the meaning of a correlation than a Covariance, it is covariances that will prove to be the
fundamental variables of our analysis. An analogy here is that variance
rates were the fundamental variables for the EWMA and GARCH
methods in Chapter 5, even though volatilities are easier to understand.
Correlation vs. Dependence
Two variables are defined as statistically independent if knowledge about
one of them does not affect the probability distribution for the other.
Formally, V1 and V2 and independent if
f(V2|V1=x)
= f(V2)
for all x, where f ( • ) denotes the probability density function.
If the coefficient of correlation between two variables is zero, does this
mean that there is no dependence between the variables? The answer is
no. We can illustrate this with a simple example. Suppose that there are
three equally likely values for V1.: - 1 , 0 , and + 1 . If V1 = — 1 or V1 = +1,
then V2 = 1. If V 1 = 0 , then V2 = 0. In this case there is clearly a
dependence between V1 and V2. If we observe the value of V1 we know
the value of V2. Also, a knowledge of the value of V2 will cause us to
145
Correlations and Copulas
(c)
Figure 6.1 Examples of ways in which V2 can depend on V1.
change our probability distribution for V1. However, the coefficient of
correlation between V1 and V2 is zero.
This example emphasizes the point that the coefficient of correlation
measures one particular type of dependence between two variables. This
is linear dependence. There are many other ways in which two variables
can be related. We can characterize the nature of the dependence between
V1 and V2 by plotting E(V2) against V1. Three examples are shown in
Figure 6.1. Figure 6.1a shows linear dependence where the expected value
of V2 depends linearly on V1. Figure 6.1b shows a V-shaped relationship
between the expected value of V2 and V1. (This is similar to the example
we have just considered; a symmetrical V-shaped relationship, however
strong, leads to zero coefficient of correlation.) Figure 6.1c shows a type
of dependence that is often seen when V1 and V2 are percentage changes
in financial variables. For the values of V1 normally encountered, there is
very little relation between V1 and V2. However, extreme values of V1 tend
146
Chapter 6
to lead to extreme values of V2. To quote one commentator: "During a
crisis the correlations all go to one."
6.2 MONITORING CORRELATION
Chapter 5 explained how EWMA and GARCH methods can be developed to monitor the variance rate of a variable. Similar approaches can
be used to monitor the Covariance rate between two variables. The
variance rate per day of a variable is the variance of daily returns.
Similarly the Covariance rate per day between two variables is defined
as the Covariance between the daily returns of the variables.
Suppose that
and
are the values of two variables X and Y at the
end of day i. The returns on the variables on day i are
The Covariance rate between X and Y on day n is, from equation (6.2),
covn = E(xnyn) - E(xn)E(yn)
In Section 5.5 we explained that risk managers assume that expected daily
returns are zero when the variance rate per day is calculated. They do the
same when calculating the Covariance rate per day. This means that the
Covariance rate per day between X and Y on day n is simply
Using equal weighting for the last m observations on
and
A similar weighting scheme for variances gives an estimate for the
variance rate on day n for variable X as
and for variable Y as
Correlations and Copulas
147
The correlation estimate on day n is
Using EWMA
Most risk managers would agree that observations from far back in the
past should not have as much weight as recent observations. In Chapter 5
we discussed the use of the EWMA model for variances. We saw that it
leads to weights that decline exponentially as we move back through time.
A similar weighting scheme can be used for covariances. The formula for
updating a Covariance estimate in the EWMA model is, similarly to
equation (5.8),
A similar analysis to that presented for the EWMA volatility model shows
that the weight given to
declines as i increases (i.e., as we move
back through time). The lower the value of the greater the weight that
is given to recent observations.
Example 6.1
Suppose that = 0.95 and that the estimate of the correlation between two
variables X and Y on day n — 1 is 0.6. Suppose further that the estimate of the
volatilities for X and Y on day n — 1 are 1% and 2%, respectively. From the
relationship between correlation and Covariance, the estimate of the Covariance rate between X and Y on day n — 1 is
0.6 x 0.01 x 0.02 = 0.00012
Suppose that the percentage changes in X and Y on day n — 1 are 0.5% and
2.5%, respectively. The variance rates and Covariance rate for day n would be
updated as follows:
= 0.95 x 0.012 + 0.05 x 0.0052 = 0.00009625
= 0.95 x 0.022 + 0.05 x 0.0252 = 0.00041125
= 0.95 x 0.00012 + 0.05 x 0.005 x 0.025 = 0.00012025
The new volatility of X is
= 0.981 % and the new volatility of Y
is
= 2.028%. The new correlation between X and Y is
148
Chapter 6
Using GARCH
GARCH models can also be used for updating Covariance rate estimates
and forecasting the future level of Covariance rates. For example, the
GARCH(1,1) model for updating a Covariance rate between X and Y is
This formula, like its counterpart in equation (5.10) for updating
variances, gives some weight to a long-run average Covariance, some
to the most recent Covariance estimate, and some to the most recent
observation on Covariance (which is
). The long-term average
Covariance rate is
Formulas similar to those in
equations (5.14) and (5.15) can be developed for forecasting future
Covariance rates and calculating the average Covariance rate during the
life of an option.
Consistency Condition for Covariances
Once variance and Covariance rates have been calculated for a set of
market variables, a variance-covariance matrix can be constructed. When
i j, the (i, j)th element of this matrix shows the Covariance rate
between the variables i and j; when i = j, it shows the variance rate of
variable i.
Not all variance-covariance matrices are internally consistent. The
condition for an N x N variance-covariance matrix
to be internally
consistent is
for all N x 1 vectors
where
is the transpose of w. A matrix that
satisfies this property is known as Positive-semidefinite.
To understand why the condition in equation (6.4) must hold, suppose
that
is
The expression
is the variance rate of
where is the value of variable i. As such, it
cannot be negative.
To ensure that a Positive-semidefinite matrix is produced, variances and
covariances should be calculated consistently. For example, if variance
rates are calculated by giving equal weight to the last m data items, the
same should be done for Covariance rates. If variance rates are updated
using an EWMA model with = 0.94, the same should be done for
Covariance rates. Multivariate GARCH models, where variance rates
Correlations and Copulas
149
and Covariance rates for a set of variables are updated in a consistent way,
can also be developed.1
An example of a variance-covariance matrix that is not internally
consistent is
The variance of each variable is 1.0 and so the covariances are also
coefficients of correlation in this case. The first variable is highly correlated
with the third variable, and the second variable is also highly correlated
with the third variable. However, there is no correlation at all between the
first and second variables. This seems strange. When we set
equal to
(1,1,-1), we find that the condition in equation (6.4) is not satisfied,
proving that the matrix is not Positive-semidefinite.2
Variance-covariance matrices that are calculated in a consistent way
from observations on the underlying variables are always Positive-semidefinite. For example, if we have 500 days of data on three different
variables and use it to calculate a variance-covariance matrix using
EWMA with = 0.94, it will be Positive-semidefinite. If we make a small
change to the matrix (e.g., for the purposes of doing a sensitivity
analysis), it is likely that the matrix will remain Positive-semidefinite.
However, if we do the same thing for observations on 1000 variables,
we have to be much more careful. The 1000 x 1000 matrix that we
calculate from the 500 days of data is Positive-semidefinite, but if we
make an arbitrary small change to the matrix it is quite likely that it will
no longer be Positive-semidefinite.
6.3 MULTIVARIATE NORMAL DISTRIBUTIONS
Multivariate normal distributions are well understood and relatively easy
to deal with. As we will explain in the next section, they can be useful
1
See R. Engle and J. Mezrich, "GARCH for Groups," Risk, August 1996, 36-40, for a
discussion of alternative approaches.
2
It can be shown that the condition for a 3 x 3 matrix of correlations to be internally
consistent is
where
is the coefficient of correlation between variables i and j.
150
Chapter 6
tools for specifying the correlation structure between variables—even
when the distributions of the variables are not normal.
We start by considering a bivariate normal distribution, where there are
only two variables, V1 and V2. Suppose that we know V1 has some
value
Conditional on this, the value of V2 is normal with mean
and standard deviation
Here
and
are the unconditional means of V1 and V2;
and
are
their unconditional standard deviations; and is the coefficient of correlation between V1 and V2. Note that the expected value of V2 conditional on V1 is linearly dependent on the value of V1. This corresponds to
Figure 6.1a. Also the standard deviation of V2 conditional on the value of
V1 is the same for all values of V1.
Generating Random Samples from Normal Distributions
Most programming languages have routines for sampling a random
number between 0 and 1 and many have routines for sampling from a
normal distribution.3 If no routine for sampling from a standardized
normal distribution is readily available, an approximate random sample
can be calculated as
where the Ri (1 i
12) are independent random numbers between 0
and 1, and is the required sample. This approximation is satisfactory for
most purposes.
When two correlated samples and from bivariate distributions are
required, an appropriate procedure is as follows. Independent samples z1
and z2 from a univariate standardized normal distribution are obtained as
just described. The required samples
and
are then calculated as
follows:
where
3
is the coefficient of correlation.
In Excel the instruction =NORMSINV(RAND()) gives a random sample from a
normal distribution.
Correlations and Copulas
151
Consider next the situation where we require n correlated samples from
normal distributions and the coefficient of correlation between sample i
and sample j is
We first sample n independent variables (1 i n)
from univariate standardized normal distributions. The required samples
are (1 i n), where
and the
are parameters chosen to give the correct variances and
correlations for the
For 1 j < i, we have
and, for all j < i,
The first sample, , is set equal to z\. These equations for the
can be
solved so that is calculated from z1 and z2, is calculated from z1, z2 and
z3, and so on. The procedure is known as the Cholesky decomposition.
If we find ourselves trying to take the square root of a negative number
when using the Cholesky decomposition, the variance-covariance matrix
assumed for the variables is not internally consistent. As explained in
Section 6.2, this is equivalent to saying that the matrix is not Positivesemidefinite.
Factor Models
Sometimes the correlations between normally distributed variables are
defined using a factor model. Suppose that U1,U2,... ,UN have standard
normal distributions (i.e., normal distributions with mean 0 and standard
deviation 1). In a one-factor model each Ui has a component dependent on
a common factor F and a component that is uncorrelated with the other
variables. Formally,
where F and the Zi have a standard normal distributions and ai is a
constant between —1 and +1. The Zi are uncorrelated with each other
and uncorrelated with F. In this model all the correlation between Ui and
Vj arises from their dependence on the common factor F. The coefficient
of correlation between Ui and Uj is
152
Chapter 6
The advantage of a one-factor model is that it imposes some structure
on the correlations. Without assuming a factor model the number of
correlations that have to be estimated for the N variables is N(N — l)/2.
With the one-factor model we need only estimate N parameters:
An example of a one-factor model from the world of
investments is the capital asset pricing model, where the return on a stock
has a component dependent on the return from the market and an
idiosyncratic (nonsystematic) component that is independent of the return
on other stocks (see Section 1.1).
The one-factor model can be extended to a two-, three-, or M-factor
model. In the M-factor model,
The factors F1, F2,..., FM have uncorrelated standard normal distributions and the Zi are uncorrelated both with each other and with the F's.
In this case the correlation between Ui and Uj is
6.4 COPULAS
Consider two correlated variables V\ and V 2 .The marginal distribution of
V1 (sometimes also referred to as the unconditional distribution) is its
distribution assuming we know nothing about V2; similarly, the marginal
distribution of V2 is its distribution assuming we know nothing about V1.
Suppose we have estimated the marginal distributions of V1 and V2. How
can we make an assumption about the correlation structure between the
two variables to define their joint distribution?
If the marginal distributions of V1 and V2 are normal, an assumption
that is convenient and easy to work with is that the joint distribution of
the variables is bivariate normal. 4 Similar assumptions are possible for
some other marginal distributions. But often there is no natural way of
defining a correlation structure between two marginal distributions. This
is where copulas come in.
4
Although this is a convenient assumption it is not the only one that can be made. There
are many other ways in which two normally distributed variables can be dependent on
each other. See, for example, Problem 6.11.
Correlations and Copulas
153
(a)
Figure 6.2
(b)
Triangular distributions for (a) V1 and (b) V2.
As an example of the application of copulas, suppose that the marginal
distributions of V1 and V2 are the triangular probability density functions
shown in Figure 6.2. Both variables have values between 0 and 1. The
density function for V1 peaks at 0.2, and the density function for V2 peaks
at 0.5. For both density functions, the maximum height is 2.0. To use what
is known as a Gaussian copula, we map V1 and V2 into new variables U1 and
U2 that have standard normal distributions. (A standard normal distribution is a normal distribution with mean 0 and standard deviation 1.) The
mapping is effected on a percentile-to-percentile basis. The 1-percentile
point of the V1 distribution is mapped to the 1-percentile point of the U1
distribution; the 10-percentile point of the V1 distribution is mapped to the
10-percentile point of the U1 distribution; and so on. The variable V2 is
mapped into U2 in a similar way. Table 6.1 shows how values of V1 are
Table 6.1 Mapping of V1, which has the triangular
distribution in Figure 6.2a, to U1, which has a standard
normal distribution.
V1 value
Percentile
of distribution
U1 value
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
5.00
20.00
38.75
55.00
68.75
80.00
88.75
95.00
98.75
-1.64
-0.84
-0.29
0.13
0.49
0.84
1.21
1.64
2.24
Chapter 6
154
Table 6.2 Mapping of V2, which has the triangular
distribution in Figure 6.2b, to U2, which has a standard
normal distribution.
V2 value
Percentile
of distribution
U2 value
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
2.00
8.00
18.00
32.00
50.00
68.00
82.00
92.00
98.00
-2.05
-1.41
-0.92
-0.47
0.00
0.47
0.92
1.41
2.05
mapped into values of U1 and Table 6.2 how values of V2 are mapped into
values of U2. Consider the V1 = 0 . 1 calculation in Table 6.1. The cumulative probability that V1 is less than 0.1 is (by calculating areas of triangles)
0.5 x 0.1 x 1 = 0.05, or 5%. The value 0.1 for V1 therefore gets mapped to
the 5-percentile point of the standard normal distribution. This is —1.64.5
The variables U1 and U2 have normal distributions. We assume that
they are jointly bivariate normal. This in turn implies a joint distribution
and a correlation structure between V1 and V2. The essence of copulas is
therefore that, instead of defining a correlation structure between V1 and
V2 directly, we do so indirectly. We map V1 and V2 into other variables
which have "well-behaved" distributions and for which it is easy to define
a correlation structure.
The way in which a copula defines a joint distribution is illustrated in
Figure 6.3. Let us assume that the correlation between U1 and U2 is 0.5.
The joint cumulative probability distribution between V1 and V2 is shown
in Table 6.3. To illustrate the calculations, consider the first one where we
are calculating the probability that V1 < 0.1 and V2 < 0.1. From Tables 6.1
and 6.2, this is the same as the probability that U1 < —1.64 and
U2 < -2.05. From the cumulative bivariate normal distribution, this is
0.006 when = 0.5.6 (The probability would be only 0.02 x 0.05 = 0.001
if = 0 . )
5
It can be calculated using Excel: NORMSINV(0.05) = -1.64.
An Excel function for calculating the cumulative bivariate normal distribution can be
found on the author's website: www.rotman.utoronto.ca/~hull.
6
155
Correlations and Copulas
One-to-one
mappings
Correlation assumption
Figure 6.3
The way in which a copula model defines a joint distribution.
Table 6.3 Cumulative joint probability distribution for V1 and V2 in a
Gaussian copula model. Correlation parameter =0.5. Table shows the joint
probability that V1 and V2 are less than the specified values.
V1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
V2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.006
0.013
0.017
0.019
0.019
0.020
0.020
0.020
0.020
0.017
0.043
0.061
0.071
0.076
0.078
0.079
0.080
0.080
0.028
0.081
0.124
0.149
0.164
0.173
0.177
0.179
0.180
0.037
0.120
0.197
0.248
0.281
0.301
0.312
0.318
0.320
0.044
0.156
0.273
0.358
0.417
0.456
0.481
0.494
0.499
0.048
0.181
0.331
0.449
0.537
0.600
0.642
0.667
0.678
0.049
0.193
0.364
0.505
0.616
0.701
0.760
0.798
0.816
0.050
0.198
0.381
0.535
0.663
0.763
0.837
0.887
0.913
0.050
0.200
0.387
0.548
0.683
0.793
0.877
0.936
0.970
156
Chapter 6
The correlation between U1 and U2 is referred to as the copula correlation. This is not, in general, the same as the correlation between V1 and
V2. Since U1 and U2 are bivariate normal, the conditional mean of U2 is
linearly dependent on U1 and the conditional standard deviation of U2 is
constant (as discussed in Section 6.3). However, a similar result does not
in general apply to V1 and V2.
Expressing the Approach Algebraically
For a more formal description of the Gaussian copula approach, suppose
that F1 and F2 are the cumulative marginal probability distributions of V1
and V2. We map V1 = v1 to U1 = u1 and V2 = v2 to U2 = u2, where
and N is the cumulative normal distribution function. This means that
and
The variables U1 and U2 are then assumed to be bivariate normal. The
key property of a copula model is that it preserves the marginal distributions of V1 and V2 (however unusual these may be) while defining a
correlation structure between them.
Other Copulas
The Gaussian copula is just one copula that can be used to define a
correlation structure between V1 and V2. There are many other copulas
leading to many other correlation structures. One that is sometimes used
is the Student t-copula. This works in the same way as the Gaussian
copula except that the variables U1 and U2 are assumed to have a
bivariate Student t-distribution. To sample from a bivariate Student
t-distribution with f degrees of freedom and correlation
we proceed
as follows:
1. Sample from the inverse chi-square distribution to get a value (In
Excel, the CHIINV function can be used. The first argument is
RAND() and the second is f.)
2. Sample from a bivariate normal distribution with correlation as
described in Section 6.3.
3. Multiply the normally distributed samples by
Correlations and Copulas
157
Figure 6.4 shows plots of 5000 random samples from a bivariate normal,
while Figure 6.5 does the same for the bivariate Student t. The correlation
parameter is 0.5 and the number of degrees of freedom for the Student t
is 4. Define a "tail value" of a distribution as a value in the left or right
1 % tail of the distribution. There is a tail value for the normal distribution when the variable is greater than 2.33 or less than —2.33. Similarly
there is a tail value in the t-distribution when the value of the variable is
greater than 3.75 or less than -3.75. Vertical and horizontal lines in the
figures indicate when tail values occur. The figures illustrate that it is more
common for both variables to have tail values in the bivariate t-distribution than in the bivariate normal distribution. To put this another way,
the tail correlation is higher in a bivariate t-distribution that in a bivariate
normal distribution. We made the point earlier that correlations between
market variables tend to increase in extreme market conditions so that
Figure 6.1c is sometimes a better description of the correlation structure
between two variables than Figure 6.1a. This has led some researchers to
argue that the Student t-copula provides a better description of the joint
behavior of market variables than the Gaussian copula.
Figure 6.4
5000 random samples from a bivariate normal distribution.
158
Figure 6.5
Chapter 6
5000 random samples from a bivariate Student t-distribution.
Multivariate Copulas
Copulas can be used to define a correlation structure between more than
two variables. The simplest example of this is the multivariate Gaussian
copula. Suppose that there are N variables, V1 V2,..., VN, and that we
know the marginal distribution of each variable. For each i (1 i N), we
transform Vi into Ui where Ui has a standard normal distribution (the
transformation is effected on a percentile-to-percentile basis as above). We
then assume that the Ui have a multivariate normal distribution.
A Factor Copula Model
In multivariate copula models, analysts often assume a factor model for
the correlation structure between the Ui. When there is only one factor,
equation (6.6) gives
Correlations and Copulas
159
where F and the Zi have standard normal distributions. The Zi are
uncorrelated with each other and uncorrelated with F. Other factor copula
models are obtained by choosing F and the Zi to have other zero-mean
unit-variance distributions. For example, if Zi is normal and F has a
Student t-distribution, we obtain a multivariate Student t-distribution
for Ui. These distributional choices affect the nature of the dependence
between the variables.
6.5 APPLICATION TO LOAN PORTFOLIOS
We now present an application of the one-factor Gaussian copula that
will prove useful in understanding the Basel II capital requirements in
Chapter 7. Consider a portfolio of N companies. Define Ti (1 i N) as
the time when company i defaults. (We assume that all companies will
default eventually—but that the default time may be a long time, perhaps
even hundreds of years, in the future.) Denote the cumulative probability
distribution of 7} by Qi.
In order to define a correlation structure between the Ti using the onefactor Gaussian copula model, we map, for each i, Ti to a variable Ui that
has a standard normal distribution on a percentile-to-percentile basis. We
assume the factor model in equation (6.8) for the correlation structure
between the Ui:
where the variables F and Zi have independent standard normal distributions. The mappings between the Ui and Ti imply
when
From equation (6.9), the probability that Ui < U conditional on the
factor value F is
Chapter 6
160
This is also
when equation (6.10) is satisfied. Hence,
To simplify matters, we suppose that the distribution Qi of time to
default is the same for all i and equal to Q. We also assume that the
copula correlation between any two names is the same and equals Since
the copula correlation between companies i and j is
this means that
the
for all i. Equation (6.11) becomes
For a large portfolio of loans, this equation provides a good estimate of
the proportion of loans in the portfolio that default by time T. We will
refer to this as the default rate.
As F decreases, the default rate increases. How bad can the default rate
become? Because F has a standard normal distribution, the probability
that F will be less than
There is therefore a probability of Y
that the default rate will be greater than
Define V(T, X) as the default rate that will not be exceeded with
probability X, so that we are X% certain that the default rate will not
exceed V(T, X). The value of V(T, X) is determined by substituting
Y = 1 — X into the above expression:
This result was first developed by Vasicek in 1987.7
Example 6.2
Suppose that a bank has lent a total of $100 million to its retail clients. The
one-year probability of default on every loan is 2% and the amount recovered
in the event of a default averages 60%. The copula correlation parameter is
7
See O. Vasicek "Probability of Loss on a Loan Portfolio," Working Paper KMV, 1987.
Vasicek's results were published in Risk in December 2002 under the title "Loan Portfolio
Value".
Correlations and Copulas
161
estimated as 0.1. In this case,
showing that we are 99.9% certain that the default rate will not be worse than
12.8%. Losses when this worst-case loss rate occur are 100 x 0.128 x (1 — 0.6),
or $5.13 million.
SUMMARY
The measure usually considered by a risk manager to describe the relationship between two variables is the Covariance rate. The daily Covariance rate
is the correlation between the daily returns on the variables multiplied by
the product of their daily volatilities. The methods for monitoring a
Covariance rate are similar to those described in Chapter 5 for monitoring
a variance rate. Either EWMA or GARCH models can be used. In
practice, risk managers need to keep track of a variance-covariance matrix
for all the variables to which they are exposed.
The marginal distribution of a variable is the unconditional distribution
of the variable. Very often an analyst is in a situation where he or she has
estimated the marginal distributions of a set of variables and wants to
make an assumption about their correlation structure. If the marginal
distributions of the variables happen to be normal, it is natural to assume
that the variables have a multivariate normal distribution. In other situations copulas are used. The marginal distributions are transformed on a
percentile-to-percentile basis to normal distributions (or to some other
distribution for which there is a multivariate counterpart). The correlation
structure between the variables of interest is then defined indirectly from an
assumed correlation structure between the transformed variables.
When many variables are involved, analysts often use a factor model.
This is a way of reducing the number of correlation estimates that have to
be made. The correlation between any two variables is assumed to derive
solely from their correlations with the factors. The default correlation
between different companies can be modeled using a factor-based Gaussian copula model of their times to default.
FURTHER READING
Cherubini, U., E. Luciano, and W. Vecchiato, Copula Methods in Finance,
Wiley, 2004.
162
Chapter 6
Demarta, S., and A.J. McNeil, "The t Copula and Related Copulas," Working
Paper, Department of Mathematics, ETH Zentrum, Zurich, Switzerland.
Engle, R. F., and J. Mezrich, "GARCH for Groups," Risk, August 1996, 36-40.
Vasicek, O. "Probability of Loss on a Loan Portfolio," Working Paper, KMV,
1987. [Published in Risk in December 2002 under the title "Loan Portfolio
Value".]
QUESTIONS AND PROBLEMS (Answers at End of Book)
6.1. If you know the correlation between two variables, what extra information
do you need to calculate the Covariance?
6.2. What is the difference between correlation and dependence? Suppose that
y = x2 and x is normally distributed with mean 0 and standard deviation 1.
What is the correlation between x and y?
6.3. What is a factor model? Why are factor models useful when defining a
correlation structure between large numbers of variables?
6.4. What is meant by a Positive-semidefinite matrix? What are the implications of a correlation matrix not being Positive-semidefinite?
6.5. Suppose that the current daily volatilities of asset A and asset B are 1.6%
and 2.5%, respectively. The prices of the assets at close of trading yesterday were $20 and $40 and the estimate of the coefficient of correlation
between the returns on the two assets made at that time was 0.25. The
parameter used in the EWMA model is 0.95. (a) Calculate the current
estimate of the Covariance between the assets. (b) On the assumption that
the prices of the assets at close of trading today are $20.50 and $40.50,
update the correlation estimate.
6.6. Suppose that the current daily volatilities of asset X and asset Y are 1.0%
and 1.2%, respectively. The prices of the assets at close of trading yesterday were $30 and $50 and the estimate of the coefficient of correlation
between the returns on the two assets made at this time was 0.50.
Correlations and volatilities are updated using a GARCH(1,1) model.
The estimates of the model's parameters are = 0.04 and = 0.94. For
the correlation, = 0.000001, and, for the volatilities, = 0.000003. If
the prices of the two assets at close of trading today are $31 and $51, how
is the correlation estimate updated?
6.7. Suppose that in Problem 5.15 the correlation between the S&P 500 Index
(measured in dollars) and the FTSE 100 Index (measured in sterling) is
0.7, the correlation between the S&P 500 Index (measured in dollars) and
the USD/GBP exchange rate is 0.3, and the daily volatility of the S&P 500
Index is 1.6%. What is the correlation between the S&P 500 Index
Correlations and Copulas
163
(measured in dollars) and the FTSE 100 Index when it is translated to
dollars? (Hint: For three variables X, Y, and Z, the Covariance between
X + Y and Z equals the Covariance between X and Z plus the Covariance
between Y and Z.)
6.8. Suppose that two variables V1 and V2 have uniform distributions where all
values between 0 and 1 are equally likely. Use a Gaussian copula to define
the correlation structure between V1 and V2 with a copula correlation of
0.3. Produce a table similar to Table 6.3 considering values of 0.25, 0.5,
and 0.75 for V1 and V2. (A spreadsheet for calculating the cumulative
bivariate normal distribution can be found on the author's website:
www.rotman.utoronto.ca/~hull.)
6.9. Assume that you have independent random samples z1, z2, and z3 from a
standard normal distribution and want to convert them to samples e1, e2,
and e3 from a trivariate normal distribution using the Cholesky decomposition. Derive three formulas expressing e1, e2, and e3 in terms of z1, z2,
and z3 and the three correlations that are needed to define the trivariate
normal distribution.
6.10. Explain what is meant by tail dependence. How can you vary tail
dependence by the choice of copula?
6.11. Suppose that the marginal distributions of V1 and V2 are standard normal
distributions but that a Student t-copula with four degrees of freedom and
a correlation parameter of 0.5 is used to define the correlation between the
variables. How would you construct a chart showing samples from the
joint distribution?
6.12. In Table 6.3 what is the probability density function of V2 conditional on
V\ < 0.1. Compare it with the unconditional distribution of V2.
6.13. What is the median of the distribution of V2 when V1 equals 0.2 in the
example in Tables 6.1 and 6.2.
6.14. Suppose that a bank has made a large number of loans of a certain type.
The total amount lent is $500 million. The one-year probability of default
on each loan is 1.5% and the loss when a default occurs is 70% of the
amount owed. The bank uses a Gaussian copula for time to default. The
copula correlation parameter is 0.2. Estimate the loss on the portfolio that
is not expected to be exceeded with a probability of 99.5%.
ASSIGNMENT QUESTIONS
6-15. Suppose that the price of gold at close of trading yesterday was $300 and
its volatility was estimated as 1.3% per day. The price of gold at the close
of trading today is $298. Suppose further that the price of silver at the
close of trading yesterday was $8, its volatility was estimated as 1.5% per
164
Chapter 6
day, and its correlation with gold was estimated as 0.8. The price of silver
at the close of trading today is unchanged at $8. Update the volatility of
gold and silver and the correlation between gold and silver using (a) the
EWMA model with =0.94, and (b) the GARCH(1,1) model with
= 0.000002, = 0.04, and = 0.94. In practice, is the
parameter
likely to be the same for gold and silver?
6.16. The probability density function for an exponential distribution is
where x is the value of the variable and A. is a parameter. The cumulative
probability distribution is
Suppose that two variables V1 and V2
have exponential distributions with
of 1.0 and 2.0, respectively. Use a
Gaussian copula to define the correlation structure between V1 and V2 with
a copula correlation of —0.2. Produce a table similar to Table 6.3 using
values of 0.25, 0.5, 0.75, 1, 1.25, 1.5 for V1 and V2. (A spreadsheet for
calculating the cumulative bivariate normal distribution can be found on
the author's website: www.rotman.utoronto.ca/~hull.
6.17. Create an Excel spreadsheet to produce a chart similar to Figure 6.5
showing samples from a bivariate Student t-distribution with four degrees
of freedom where the correlation is 0.5. Next suppose that the marginal
distributions of V1 and V2 are Student t with four degrees of freedom but
that a Gaussian copula with a copula correlation parameter of 0.5 is used
to define the correlation between the two variables. Construct a chart
showing samples from the joint distribution. Compare the two charts you
have produced.
6.18. Suppose that a bank has made a large number loans of a certain type. The
one-year probability of default on each loan is 1.2%. The bank uses a
Gaussian copula for time to default. It is interested in estimating a
"99.97% worst case" for the percentage of loans that default on the
portfolio. Show how this varies with the copula correlation.
Bank Regulation
and Basel II
An important objective of governments is to provide a stable economic
environment for private individuals and businesses. One way they do this
is by providing a reliable banking system where bank failures are rare and
depositors are protected. Shortly after the disastrous crash of 1929, the
United States took a number of steps to increase confidence in the
banking system and protect depositors. It created the Federal Deposit
Insurance Corporation (FDIC) to provide safeguards to depositors in the
event of a failure by a bank. It also passed the famous Glass-Steagall Act
that prevented deposit-taking commercial banks from engaging in investment banking activities.
Deposit insurance continues to exist in the United States and many
other countries today. However, many of the provisions of the GlassSteagall Act in the United States have now been repealed. There has been
a trend worldwide toward the development of progressively more complicated rules on the capital that banks are required to keep. This is because,
as shown in Section 1.3, the ability of a bank to absorb unexpected losses
is critically dependent on the amount of equity and other forms of capital
held. In this chapter we review the evolution of the regulation of bank
capital from the 1980s and explain the new Basel II capital requirements,
which are scheduled to be implemented starting in 2007.
It is widely accepted that the capital a financial institution requires
should cover the difference between expected losses over some time horizon
and "worst-case losses" over the same time horizon. The worst-case loss is
166
Chapter 7
Figure 7.1
The loss probability density function and the capital
required by a financial institution.
the loss that is not expected to be exceeded with some high degree of
confidence. The high degree of confidence might be 99% or 99.9%. The
idea here is that expected losses are usually covered by the way a financial
institution prices its products. For example, the interest charged by a bank
is designed to recover expected loan losses. Capital is a cushion to protect
the bank from an extremely unfavorable outcome. This is illustrated in
Figure 7.1.
Banks compete in some financial markets with securities firms and
insurance companies. These types of financial institutions are often
subject to different regulations from banks. In the United States securities
firms are regulated by the Securities and Exchange Commission (SEC),
and insurance companies are regulated at the state level with national
guidelines being set by the National Association of Insurance Commissioners (NAIC). However, the regulators of all financial institutions face
similar problems. Bank regulators want to protect depositors and ensure
a stable financial system; insurance regulators want to protect policyholders from defaults by insurance companies and ensure that the public
has confidence in the insurance industry; securities regulators want to
protect the clients of brokers from defaults and ensure that markets
operate smoothly. In some instances the three types of regulators find
themselves specifying capital for the same financial instruments. If they
do not do this in the same way, there is liable to be what is known as
regulatory arbitrage, with risks being shifted to those financial institutions
that are required to carry least capital for the financial instruments.
Bank Regulation and Basel II
167
There have been some signs of convergence in the regulation of
financial institutions. Insurance regulators and securities regulators are
adopting similar approaches to bank regulators in prescribing minimum
levels for capital. In the European Union the Capital Requirements
Directive (CRD) legislation will require the regulatory capital for securities firms to be calculated in a similar way to that for banks. Another
initiative by the European Union, Solvency II, is likely to lead to the
capital for insurance companies in Europe being calculated in a broadly
similar way to that for banks.
Bank regulators are in many ways taking the lead in developing a
methodology for setting capital requirements for financial institutions.
The regulation of banks is based on international standards, whereas the
regulation of other types of financial institutions varies more from
country to country. For this reason the regulation of banks will be the
main focus of this chapter.
7.1 REASONS FOR REGULATING BANK CAPITAL
It is tempting to argue as follows: "Bank regulation is unnecessary. Even
if there were no regulations, banks would manage their risks prudently
and would strive to keep a level of capital that is commensurate with the
risks they are taking." Unfortunately, history does not altogether support
this view. There is little doubt that regulation has played an important
role in increasing bank capital, making banks more aware of the risks
they are taking.
If markets operated totally without government intervention, banks
that took risks by keeping low levels of equity capital would find it
difficult to attract deposits and might experience a "run on deposits",
where large numbers of depositors try to withdraw funds at the same
time. As mentioned earlier, most governments do provide some form of
deposit insurance because they want depositors to have confidence that
their money is safe. However, the existence of deposit insurance has the
effect of encouraging banks to reduce equity capital (thereby increasing
expected return on equity) because they no longer have to worry about
depositors losing confidence.1 From the government's perspective there is
therefore a risk that the existence of deposit insurance leads to more bank
. 1 This is an example of what insurance companies term moral hazard. The existence of an
insurance contract changes the behavior of the insured party. We will discuss moral
hazard further in Chapter 14.
Chapter 7
168
Business Snapshot 7.1
Systemic Risk
Systemic risk is the risk that a default by one financial institution will create a
"ripple effect" that leads to defaults by other financial institutions and
threatens the stability of the financial system. The financial system has survived
defaults such as Drexel in 1990 and Barings in 1995 very well, but regulators
continue to be concerned. There are huge numbers of over-the-counter transactions between banks. If Bank A fails, Bank B may take a huge loss on the
transactions it has with Bank A. This in turn could lead to Bank B failing,
Bank C that has many outstanding transactions with both Bank A and Bank
B might then take a large loss and experience severe financial difficulties, and
so on.
failures and an increase in the cost of deposit insurance programs. As a
result governments have found it necessary to combine deposit insurance
with regulations on the capital banks must hold. In addition, governments are concerned about what is termed systemic risk. This is discussed
in Business Snapshot 7.1.
7.2 PRE-1988
Prior to 1988 bank regulators in different countries tended to regulate
bank capital by setting minimum levels for the ratio of capital to total
assets. However, definitions of capital and the ratios considered acceptable varied from country to country. Some countries enforced their
regulations more diligently than others. Banks were competing globally
and a bank operating in a country where capital regulations were slack
was considered to have a competitive edge over one operating in a country
with tighter more strictly enforced capital regulations. In addition the
huge exposures of the major international banks to less developed
countries such as Mexico, Brazil, and Argentina and the accounting
games sometimes used to manage those exposures were starting to raise
questions about the adequacy of capital levels.
Another problem was that the types of transactions entered into by
banks were becoming more complicated. The over-the-counter derivatives
market for products such as interest rate swaps, currency swaps, and
foreign exchange options was growing fast. These contracts increase the
credit risks being taken by a bank. Consider, for example, an interest rate
swap. If the counterparty in an interest rate swap transaction defaults
Bank Regulation and Basel II
169
when the swap has a positive value to the bank and a negative value to the
counterparty, the bank loses money. Many of these newer transactions
were "off balance sheet". This means that they had no effect on the level of
assets reported by a bank. As a result, they had no effect on the amount of
capital the bank was required to keep. It became apparent to regulators
that total assets was no longer a good indicator of the total risks being
taken. A more sophisticated approach than that of setting minimum levels
for the ratio of capital to total balance sheet assets was needed.
These problems led supervisory authorities for Belgium, Canada,
France, Germany, Italy, Japan, Luxembourg, the Netherlands, Sweden,
Switzerland, the United Kingdom, and the United States to form the
Basel Committee on Banking Supervision. They met regularly in Basel,
Switzerland, under the patronage of the Bank for International Settlements. The first major result of these meetings was a document entitled
"International Convergence of Capital Measurement and Capital Standards". This was referred to as "The 1988 BIS Accord" or just "The
Accord". More recently it has come to be known as Basel I.
7.3 THE 1988 BIS ACCORD
The 1988 BIS Accord was the first attempt to set international risk-based
standards for capital adequacy. It has been subject to much criticism as
being too simple and somewhat arbitrary. In fact, the Accord was a huge
achievement. It was signed by all 12 members of the Basel Committee and
paved the way for significant increases in the resources banks devote to
measuring, understanding, and managing risks.
The BIS Accord defined two minimum standards for meeting acceptable
capital adequacy requirements. The first standard was similar to that
existing prior to 1988 and required banks to have an assets-to-capital
multiple of at most 20. The second standard introduced what became
known as the Cooke ratio. For most banks there was no problem in
satisfying the capital multiple rule. The Cooke ratio was the key regulatory
requirement.
The Cooke Ratio
In calculating the Cooke ratio both On-balance-sheet and off-balancesheet items are considered. They are used to calculate what is known as
the bank's total risk-weighted assets (also sometimes referred to as the
risk-weighted amount). It is a measure of the bank's total credit exposure.
170
Chapter 7
Table 7.1
Risk weight
Risk weights for On-balance-sheet items.
Asset category
(%)
0
20
50
100
Cash, gold bullion, claims on OECD governments
such as Treasury bonds or insured residential
mortgages
Claims on OECD banks and OECD public sector
entities such as securities issued by US
government agencies or claims on municipalities
Uninsured residential mortgage loans
All other claims, such as corporate bonds and
less-developed country debt, claims on nonOECD banks, real estate, premises, plant, and
equipment
Consider first On-balance-sheet items. Each On-balance-sheet item is
assigned a risk weight reflecting its risk. A sample of the risk weights
specified in the Accord are shown in Table 7.1. Cash and securities issued
by OECD governments are considered to have virtually zero risk and
have a risk weight of zero. Loans to corporations have a risk weight of
100%. Loans to OECD banks and government agencies have a risk
weight of 20%. Uninsured residential mortgages have a risk weight of
50%. The total risk-weighted assets for N On-balance-sheet items equals
where Li is the principal amount of the ith item and
is its risk weight.
Example 7.1
The assets of a bank consist of $100 million of corporate loans, $10 million of
OECD government bonds, and $50 million of residential mortgages. The total
of risk-weighted assets is
1.0 x 100 + 0.0 x 10 + 0.5 x 50 = 125
or $125 million.
Off-balance-sheet items are expressed as a credit equivalent amountLoosely speaking, the credit equivalent amount is the loan principal that
is considered to have the same credit risk. For nonderivative instruments
the credit equivalent amount is calculated by applying a conversion factor
Bank Regulation and Basel II
171
to the principal amount of the instrument. Instruments that from a credit
perspective are considered to be similar to loans, such as bankers'
acceptances, have a conversion factor of 100%. Others, such as note
issuance facilities (where a bank agrees that a company can issue shortterm paper on pre-agreed terms in the future), have lower conversion
factors.
For an over-the-counter derivative, such as an interest rate swap or a
forward contract, the credit equivalent amount is calculated as
where V is the current value of the derivative, a is an add-on factor,
and L is the principal amount. The first term in equation (7.1) is the
current exposure. The add-on factor is an allowance for the possibility
of the exposure increasing in the future. The add-on factors are shown
in Table 7.2.
Example 7.2
A bank has entered into a $100 million interest rate swap with a remaining life
of 4 years. The current value of the swap is $2.0 million. In this case the addon amount is 0.5% of the principal, so that the credit equivalent amount is
$2.0 million plus $0.5 million, or $2.5 million.
The credit equivalent amount for an off-balance-sheet item is multiplied
by the risk weight for the counterparty in order to calculate the riskweighted assets. The risk weights for off-balance-sheet items are similar to
those in Table 7.1 except that the risk weight for a corporation is 0.5
rather than 1.0 when off-balance-sheet items are considered.
Example 7.3
Consider again the bank in Example 7.2. If the interest rate swap is with a
corporation, the risk-weighted assets are 2.5 x 0.5, or $1.25 million. If it is
with an OECD bank, the risk-weighted assets are 2.5 x 0.2, or $0.5 million.
Table 7.2 Add-on factors (as a percentage of principal) for derivatives.
Remaining
maturity
(years)
Interest
rate
Exchange rate
and gold
Equity
Precious metals
except gold
Other
commodities
<1
1 to 5
>5
0.0
0.5
1.0
5.0
7.5
6.0
8.0
10.0
7.0
7.0
8.0
10.0
12.0
15.0
1.5
172
Chapter 7
Putting all this together, the total risk-weighted assets for a bank with
N On-balance-sheet items and M off-balance-sheet items is
Here, Li is the principal of the ith On-balance-sheet item and
weight for the counterparty;
jth off-balance-sheet item and
is its risk
is the credit equivalent amount for the
is the risk weight for the counterparty.
Capital Requirement
The Accord required banks to keep capital equal to at least 8% of the
risk-weighted assets. The capital had two components:
1. Tier 1 Capital. This consists of items such as equity, noncumulative
perpetual preferred stock2 less goodwill.
2. Tier 2 Capital. This is sometimes referred to as Supplementary
Capital. It includes instruments such as cumulative perpetual preferred
stock,3 certain types of 99-year debenture issues, and subordinated debt
with an original life of more than five years.
At least 50% of the required capital (i.e., 4% of the risk-weighted assets)
must be in Tier 1. The Basel Committee updated its definition of
instruments that are eligible for Tier 1 capital in a 1998 press release.
7.4 THE G-30 POLICY RECOMMENDATIONS
In 1993 a working group consisting of end users, dealers, academics,
accountants, and lawyers involved in derivatives published a report that
contained 20 risk management recommendations for dealers and end
users of derivatives and four recommendations for legislators, regulators,
and supervisors. The report was based on a detailed survey of 80 dealers
and 72 end users worldwide. The survey involved both questionnaires and
in-depth interviews. The report is not a regulatory document, but it has
been very influential in the development of risk management practices.
2
Noncumulative perpetual preferred stock is preferred stock lasting forever where there
is a predetermined dividend rate. Unpaid dividends do not cumulate (i.e., the dividends
for one year are not carried forward to the next year).
3
In cumulative preferred stock unpaid dividends cumulate. Any backlog of dividends
must be paid before dividends are paid on the common stock.
Bank Regulation and Basel II
173
A brief summary of the important recommendations is as follows:
1. A company's policies on risk management should be clearly
defined and approved by senior management, ideally at the board
of directors level. Managers at all levels should enforce the policies.
2. Derivatives positions should be marked to market at least once
a day.
3. Derivatives dealers should measure market risk using a consistent
measure such as value at risk. (This will be discussed further in
Chapter 8.) Limits to the market risks that are taken should be set.
4. Derivatives dealers should carry out stress tests to determine
potential losses under extreme market conditions.
5. The risk management function should be set up so that it is
independent from the trading operation.
6. Credit exposures arising from derivatives trading should be assessed
based on the current replacement value of existing positions and
potential future replacement costs.
7. Credit exposures to a counterparty should be aggregated in a way
that reflects enforceable netting agreements. (We talk about netting
in the next section.)
8. The individuals responsible for setting credit limits should be
independent of those involved in trading.
9. Dealers and end users should assess carefully both the costs and
benefits of credit risk mitigation techniques such as Collateralization and downgrade triggers. In particular, they should assess their
own capacity and that of their counterparties to meet the cash flow
requirement of downgrade triggers. (Credit mitigation techniques
are discussed in Chapter 12.)
10. Only individuals with the appropriate skills and experience should
be allowed to have responsibility for trading derivatives, supervising the trading, carrying out back-office functions in relation to
the trading, etc.
11. There should be adequate systems in place for data capture,
processing, settlement, and management reporting.
12. Dealers and end users should account for the derivatives transactions used to manage risks so as to achieve a consistency of
income recognition treatment between those instruments and the
risks being managed.
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Chapter 7
7.5 NETTING
The word netting refers to a clause in over-the-counter contracts which
states that if a counterparty defaults on one contract it has with a
financial institution then it must default on all outstanding contracts with
that financial institution.
Netting can have the effect of substantially reducing credit risk.
Consider a bank that has three swap contracts outstanding with a
particular counterparty. The contracts are worth +$24 million,
—$17 million, and +$8 million to the bank. Suppose that the counterparty experiences financial difficulties and defaults on its outstanding
obligations. To the counterparty, the three contracts have values of
—$24 million, +$17 million, and —$8 million, respectively. Without
netting, the counterparty would default on the first contract, keep the
second contract, and default on the third contract. The loss to the bank
would be $32 (= 24 + 8) million. With netting, the counterparty is
required to default on the second contract as well. The loss to bank is
then $15 (= 24 - 17 + 8) million.4
Suppose that a financial institution has a portfolio of N derivative
contracts outstanding with a particular counterparty and that the current
value of the ith contract is Vi. Without netting, the financial institution's
exposure in the event of a default today is
With netting, it is
Without netting, the exposure is the payoff from a portfolio of options;
with netting, the exposure is the payoff from an option on a portfolio.
The 1988 Accord does not take netting into account in setting capital
requirements. From equation (7.1) the credit equivalent amount for a
portfolio of derivatives with a counterparty under the Accord is
4
Note that if the second contract had been worth —$40 million to the bank then the
counterparty is better off if it chooses not to default on its contracts with the bank.
Bank Regulation and Basel II
175
where ai is the add-on factor for the ith transaction and Li is the principal
for the ith transaction.
By 1995 netting had been successfully tested in the courts in many
jurisdictions. As a result, the 1988 Accord was modified to allow banks to
reduce their credit equivalent totals when enforceable bilateral netting
agreements were in place. The first step was to calculate the net replacement ratio, NRR. This is the ratio of the current exposure with netting to
the current exposure without netting:
The credit equivalent amount was modified to
Example 7.4
Consider the example in Table 7.3 which shows a portfolio of three derivatives
contracts that a bank has with a particular counterparty. The third column
shows the current marked-to-market values of the transactions and the fourth
column shows the add-on amount calculated from Table 7.2. The current
exposure with netting is —60 + 70 + 55 = 65. The current exposure without
netting is 0 + 70 + 55 = 125. The net replacement ratio is given by
The total of the add-on amounts,
is 5 + 75 + 3 0 = 1 1 0 . The
credit equivalent amount, when netting agreements are in place is
65 + (0.4 + 0.6 x 0.52) x 110 = 143.32. Without netting agreements, the
credit equivalent amount is 125 + 110 = 235. Suppose that the counterParty is an OECD bank so that the risk weight is 0.2. This means that
the risk-weighted assets with netting is 0.2 x 143.32 = 28.66. Without
netting, it is 0.2 x 235 = 47.
Table 7.3
Portfolio of derivatives with a particular counterparty.
Transaction
3-year interest rate swap
6-year foreign exchange forward
9-month option on a stock
Principal,
Current value,
Li
Vi
Table 7.2 add-on
amount, ai Li
1000
1000
500
-60
70
55
5
75
30
176
Chapter 7
7.6 THE 1996 AMENDMENT
In 1995 the Basel Committee issued a consultative proposal to amend
the 1988 Accord. This became known as the "1996 Amendment". It
was implemented in 1998 and was then sometimes referred to as
"BIS 98".
The 1996 Amendment requires financial institutions to hold capital to
cover their exposure to market risks as well as credit risks. The Amendment distinguishes between a bank's trading book and its banking book.
The banking book consists primarily of loans and is not usually marked
to market for managerial and accounting purposes. The trading book
consists of the myriad of different instruments that are traded by the bank
(stocks, bonds, swaps, forward contract, exotic derivatives, etc.). The
trading book is normally marked to market daily.
Under the 1996 Amendment, the credit risk capital charge in the 1988
Accord continued to apply to all On-balance-sheet and off-balance-sheet
items in the trading and banking book, except positions in the trading
book that consisted of (a) debt and equity traded securities and
(b) positions in commodities and foreign exchange. In addition there
was a market risk capital charge for all items in the trading book whether
they were on balance sheet or off balance sheet.5
The 1996 Amendment outlined a standardized approach for measuring
the capital charge for market risk. The standardized approach assigned
capital separately to each of debt securities, equity securities, foreign
exchange risk, commodities risk, and options. No account was taken of
correlations between different types of instruments. The more sophisticated
banks with well-established risk management functions were allowed to
use an "internal model-based approach" for setting market risk capital.
This involved calculating a value-at-risk measure and converting it into a
capital requirement using a formula specified in the 1996 amendment. (We
discuss value at risk and the alternative approaches companies use to
calculate it in Chapters 8, 9, and 10). Most large banks preferred to use
the internal model-based approach because it better reflected the benefits
of diversification and led to lower capital requirements.
The value-at-risk measure used by regulators for market risk is the loss
on the trading book that can be expected to occur over a 10-day period
1% of the time. Suppose that the value at risk is $1 million. This means
5
Certain nontrading book positions that are used to hedge positions in the trading book
can be included in the calculation of the market risk capital charge.
Bank Regulation and Basel II
177
that the bank is 99% confident that there will not be a loss greater than
$1 million over the next 10 days.
The market risk capital requirement for banks when they use the
internal model-based approach is calculated at any given time as
k * VaR + SRC
(7.3)
where k is a multiplicative factor, and SRC is a specific risk charge. The
value at risk, VaR, is the greater of the previous day's value at risk and
the average value at risk over the last 60 days. The minimum value for k
is 3. Higher values may be chosen by regulators for a particular bank if
tests reveal inadequacies in the bank's value-at-risk model.
The specific risk charge, SRC, is a capital charge for the idiosyncratic
risks related to individual companies. One security that gives rise to
idiosyncratic risk is a corporate bond. There are two components to the
risk of this security: interest rate risk and credit risk. The interest rate risk
is captured by the bank's market value-at-risk measure; the credit risk is
specific risk.6 The 1996 Amendment proposed standard methods for
assessing a specific risk capital charge, but allowed banks to use internal
models for arriving at a capital charge for specific risk similarly to the
way they calculate a capital charge for market risks. We discuss specific
risk and its calculation further in Chapter 12.
The total capital a bank was required to keep after the implementation
of the 1996 Amendment was (for banks adopting the internal model-based
approach) the sum of (a) credit risk capital equal to 8% of the riskweighted assets (RWA), as given by equation (7.2) and (b) market risk
capital as given by equation (7.3). For convenience, an RWA for market
risk capital is defined as 12.5 multiplied by the amount given in
equation (7.3). This means that the total capital required for credit and
market risk is given by
Total capital = 0.08 x (Credit risk RWA + Market risk RWA)
(7.4)
A bank has more flexibility in the type of capital it uses for market risk. It
can use Tier 1 or Tier 2 capital. It can also use what is termed Tier 3
capital. This consists of short-term subordinated debt with an original
maturity of at least two years that is unsecured and fully paid up.
For most banks the majority of the capital is for credit risk rather than
6
As mentioned earlier, the 1988 credit capital charge did not apply to debt securities in
the trading book under the 1996 Amendment.
178
Chapter 7
market risk. In a typical situation 70% of required capital might be for
credit risk and 30% for market risk.
7.7 BASEL II
The 1988 Basel Accord led to significant increases in the capital held by
banks over the following ten years. It deserves a great deal of credit for
improving the stability of the global banking system. However, it does
have significant weaknesses. Under the 1988 Basel Accord, all loans by a
bank to a corporation have a risk weight of 100% and require the same
amount of capital. A loan to a corporation with a AAA credit rating is
treated in the same way as one to a corporation with a B credit rating.7
Furthermore, in Basel I, there was no model of default correlation.
In June 1999 the Basel Committee proposed new rules that have
become known as Basel II. These were revised in January 2001 and April
2003. A number of quantitative impact studies (QISs) were carried out to
test the application of the new rules and the amount of capital that will be
required. A final set of rules agreed to by all members of the Basel
Committee was published in June 2004. This was updated in November
2005. Implementation of the rules is expected to begin in 2007 after a
further QIS. 8
The Basel II capital requirements apply to "internationally active"
banks. In the United States there are many small regional banks and
the US regulatory authorities have decided that Basel II will not apply to
them. (These banks will be regulated under what is termed Basel IA,
which is similar to Basel I.) It is likely that some of the larger regional
banks will voluntarily implement Basel II—perhaps to signal to their
shareholders that they manage risks in a sophisticated way. In Europe all
banks, large or small, will be regulated under Basel II. In addition, the
European Union requires the Basel II rules to be applied to securities
companies as well as banks. Basel II is based on three "pillars":
1. Minimum capital requirements
2. Supervisory review
3. Market discipline
7
Credit ratings are discussed in Chapter 11.
One point to note about the QIS studies is that they do not take account of changes
banks may choose to make to their portfolios to minimize their capital requirements once
Basel II is implemented.
8
Bank Regulation and Basel II
179
In Pillar 1, the minimum capital requirement for credit risk in the banking book is calculated in a new way that reflects the credit ratings of
counterparties. The capital requirement for market risk remains unchanged from the 1996 Amendment and there is a new capital charge
for operational risk. The general requirement in Basel I that banks hold a
total capital equal to 8% of risk-weighted assets remains unchanged. A
risk-weighted asset for operational risk is defined as 12.5 times the
calculated operational risk capital and equation (7.4) becomes
Total capital = 0.08 x
(Credit risk RWA + Market risk RWA + Operational risk RWA)
(7.5)
Pillar 2, which is concerned with the supervisory review process, allows
regulators in different countries some discretion in how rules are applied
(so that they can take account of local conditions) but seeks to achieve
overall consistency in the application of the rules. It places more
emphasis on early intervention when problems arise. Supervisors are
required to do far more than just ensure that the minimum capital
required under Basel II is held. Part of their role is to encourage banks
to develop and use better risk management techniques and to evaluate
these techniques. They should evaluate risks that are not covered by
Pillar 1 and enter into an active dialogue with banks when deficiencies
are identified.
The third pillar, market discipline, will require banks to disclose more
information about the way they allocate capital and the risks they take.
The idea here is that banks will be subjected to added pressure to make
sound risk management decisions if shareholders and potential shareholders have more information about those decisions.
7.8 CREDIT RISK CAPITAL UNDER BASEL II
For credit risk, banks will have three choices under Basel II:
1. The standardized approach
2. The foundation internal ratings based (IRB) approach
3. The advanced IRB approach
The overall structure of the calculations is similar to that under Basel I.
For an On-balance-sheet item a risk weight is applied to the principal
to calculate risk-weighted assets reflecting the creditworthiness of the
180
Chapter 7
counterparty. For off-balance-sheet items the risk weight is applied to a
credit equivalent amount. This is calculated using either credit conversion
factors or add-on amounts. The adjustments for netting are similar to
those in Basel I (see Section 7.5).
The Standardized Approach
The standardized approach is to be used by banks that are not sufficiently sophisticated (in the eyes of the regulators) to use the internal
ratings approaches. (In the United States, Basel II will apply only to the
largest banks and US regulators have decided that these banks must use
the IRB approach.) Some of the rules for determining risk weights are
summarized in Table 7.4. Comparing Table 7.4 with Table 7.1, we see
that the OECD status of a bank or a country is no longer important
under Basel II. The risk weight for a country (sovereign) exposure ranges
from 0% to 150% and the risk weight for an exposure to another bank
or a corporation ranges from 20% to 150%. In Table 7.1 OECD banks
were implicitly assumed to be lesser credit risks than corporations. An
OECD bank attracted a risk weight of 20%, while a corporation
attracted a risk weight of 100%. Table 7.4 treats banks and corporation
much more equitably. An interesting observation from Table 7.4 for a
country, corporation, or bank that wants to borrow money is that it may
be better to have no credit rating at all than a very poor credit rating!
Supervisors are allowed to apply lower risk weights (20% rather than
50%, 50% rather than 100%, and 100% rather than 150%) when
exposures are to a bank's country of incorporation or that country's
central bank.
Table 7.4 Risk weights (as a percentage of principal) for exposures to
country, banks, and corporations under Basel II's standardized approach as
a function of their ratings.
Country*
Banks**
Corporations
AAA
to
AA-
A+
0
20
20
B+
to
A-
BBB+
to
BBB-
BB+
to
BB-
to
B-
20
50
50
50
50
100
100
100
100
100
100
150
* Includes exposures to central banks of the country.
** National supervisors have options as outlined in the text.
Below
B-
Unrated
150
150
150
100
50
100
Bank Regulation and Basel II
181
For claims on banks, the rules are somewhat complicated. Instead of
using the risk weights in Table 7.4, national supervisors can choose to
base capital requirements on the rating of the country in which the bank
is incorporated. The risk weight assigned to the bank will be 20% if the
country of incorporation has a rating between AAA and A A - , 50% if it
is between A+ and A - , 100% if it is between BBB+ and B - , 150% if it
is below B-, and 100% if it is unrated. Another complication is that, if
national supervisors elect to use the rules in Table 7.4, then they can
choose to treat claims with a maturity less than three months more
favorably, so that the risk weights are 20% if the rating is between AAA+
and BBB-, 50% if it is between BB+ and B - , 150% if it is below B - ,
and 20% if it is unrated.
The standard rule for retail lending is that a risk weight of 75% be
applied. (This compares with 100% in the 1988 Accord.) When claims
are secured by a residential mortgage, the risk weight is 35%. (This
compares with 50% in the 1988 Accord.) Owing to poor historical loss
experience, the risk weight for claims secured by commercial real estate
is 100%.
Example 7.5
Suppose that the assets of a bank consist of $100 million of loans to corporations rated A, $10 million of government bonds rated AAA, and $50 million of
residential mortgages. Under the Basel II standardized approach, the total
risk-weighted assets is
0.5 x 100 + 0.0 x 10 + 0.35 x 50 = 67.5
or $67.5 million. This compares with $125 million under Basel I (see
Example 7.1).
Adjustments for Collateral
There are two ways banks can adjust risk weights for collateral. The first
is termed the simple approach and is similar to an approach used in
Basel I. The second is termed the comprehensive approach. Banks have
a choice as to which approach is used in the banking book, but they must
use the comprehensive approach to calculate capital for counterparty
credit risk in the trading book.
Under the simple approach, the risk weight of the counterparty is
replaced by the risk weight of the collateral for the part of the exposure
covered by the collateral. (The exposure is calculated after netting.) For
any exposure not covered by the collateral, the risk weight of the counterparty is used. The minimum level for the risk weight applied to the
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Chapter 7
collateral is 20%. 9 A requirement is that the collateral must be revalued at
least every six months and must be pledged for at least the life of the
exposure.
Under the comprehensive approach, banks adjust the size of their
exposure upward to allow for possible increases and adjust the value of
the collateral downward to allow for possible decreases in the value of the
collateral.10 (The adjustments depend on the volatility of the exposure and
the loan.) A new exposure equal to the excess of the adjusted exposure
over the adjusted value of the collateral is calculated and the counterparty's risk weight is applied to this exposure. The adjustments applied to
the exposure and the collateral can be calculated using rules specified in
Basel II or, with regulatory approval, using a bank's internal models.
Where netting arrangements apply, exposures and collateral are separately
netted and the adjustments made are weighted averages.
Example 7.6
Suppose that an $80 million exposure to a particular counterparty is secured
by collateral worth $70 million. The collateral consists of bonds issued by an
A-rated company. The counterparty has a rating of B+. The risk weight for
the counterparty is 150% and the risk weight for the collateral is 50%. The
risk-weighted assets applicable to the exposure using the simple approach is
therefore
0.5 x 70+1.50 x 10 = 50
or $50 million.
Consider next the comprehensive approach. Assume that the adjustment to
exposure to allow for possible future increases in the exposure is +10% and
the adjustment to the collateral to allow for possible future decreases in its
value is —15%. The new exposure is
1.1x80-0.85x70 = 28.5
or $28.5 million and a risk weight of 150% is applied to this exposure to give
risk-adjusted assets equal to $42.75 million.
The IRB Approach
The model underlying the internal ratings based (IRB) approach is the
one-factor Gaussian copula model of time to default that we discussed in
Section 6.5.
9
An exception is when the collateral consists of cash or government securities with the
currency of the collateral being the same as the currency of the exposure.
10
An adjustment to the exposure is not likely to be necessary on a loan, but is likely to be
necessary on an over-the-counter derivative. The adjustment is in addition to the add on
factor.
Bank Regulation and Basel II
Table 7.5
=
=
=
=
=
0.0
0.2
0.4
0.6
0.8
183
Dependence of WCDR on PD and
PD = 0.1%
PD = 0.5%
P D = 1%
P D = 1.5%
PD = 2.0%
0.1%
2.8%
7.1%
13.5%
23.3%
0.5%
9.1%
21.1%
38.7%
66.3%
1.0%
14.6%
31.6%
54.2%
83.6%
1.5%
18.9%
39.0%
63.8%
90.8%
2.0%
22.6%
44.9%
70.5%
94.4%
Consider a large portfolio of N loans. Define:
WCDR: The worst-case default rate during the next year that we are
99.9% certain will not be exceeded
PD: The probability of default for each loan in one year
EAD: The exposure at default on each loan (in dollars)
LGD: The loss given default. This is the proportion of the exposure
that is lost in the event of a default
Suppose that the copula correlation between each pair of obligors is
Equation (6.12) shows that
11
It follows that there is a 99.9% chance that the loss on the portfolio will
be less than N times
EAD x LGD x WCDR
It can be shown that, as a good approximation, this result can be
extended to the case where the loans have different sizes and different
default probabilities. In a general portfolio of loans, there is a
99.9% chance that the total loss will be less than the sum of
EAD x LGD x WCDR for the individual loans.12 This result is the
theoretical underpinning of the IRB approach.
Table 7.5 shows how WCDR depends on PD and
When the
correlation
is zero, WCDR = PD because in that case there is no
default correlation and the default rate in all years is the same. As
increases, WCDR increases.
11
Note that the Basel Committee publications use R, not
to denote the copula
correlation.
12
The WCDR for an individual loan is calculated by substituting the PD and
Parameter for the loan into equation (7.6).
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Chapter 7
Corporate, Sovereign, and Bank Exposures
In the case of corporate, sovereign, and bank exposures, Basel II assumes
a relationship between the correlation parameter and the probability of
default PD in equation (7.6) based on empirical research.13 The formula is
Because exp(—50) is a very small number, this formula is to all intents
and purposes
As PD increases, decreases. The reason for this inverse relationship is as
follows. As a company becomes less creditworthy, its PD increases and its
probability of default becomes more idiosyncratic and less affected by
overall market conditions.
Combining equation (7.7) with equation (7.6), we obtain the relationship between WCDR and PD in Table 7.6. We find that WCDR is, as we
would expect, an increasing function of PD. However, it does not increase
as fast as it would if were assumed to be independent of PD.
The formula for the capital required is
EAD x LGD x (WCDR - PD) x MA
(7.8)
The first three terms in this expression can be understood from our earlier
discussion. We use WCDR — PD instead of WCDR because we are
interested in providing capital for the excess of the 99.9% worst-case loss
over the expected loss. The variable MA is the maturity adjustment and is
defined as
Table 7.6 Relationship between WCDR and PD for
corporate, sovereign, and bank exposures.
PD:
0.1%
0.5%
1.0%
1.5%
2.0%
WCDR:
3.4%
9.8%
14.0%
16.9%
19.0%
13
See J. Lopez, "The Empirical Relationship Between Average asset Correlation, Firm
Probability of Default and Asset Size," Journal of Financial Intermediation, 13, 2 (2004),
265-283.
Bank Regulation and Basel II
185
where
b = [0.11852 - 0.05478 x ln(PD)] 2
and M is the maturity of the exposure.
The maturity adjustment is designed to allow for the fact that, if an
instrument lasts longer than one year, there is a one-year credit exposure
arising from a possible decline in the creditworthiness of the counterparty
as well as from a possible default by the counterparty. (Note that when
M = 1 the MA is 1.0 and has no effect.) As mentioned earlier, the riskweighted assets (RWA) are calculated as 12.5 times the capital required
RWA = 12.5 x EAD x LGD x (WCDR - PD) x MA
so that the capital is 8% of RWA.
Under the foundation IRB approach, banks supply PD, while LGD,
EAD, and M are supervisory values set by the Basel Committee. PD is
largely determined by a bank's own estimate of the creditworthiness of
the counterparty. It is subject to a floor of 0.03% for bank and corporate
exposures. LGD is set at 45% for senior claims and 75% for subordinated claims. When there is eligible collateral, in order to correspond to the
comprehensive approach that we described earlier, LGD is reduced by the
ratio of the adjusted value of the collateral to the adjusted value of the
exposure, both calculated using the comprehensive approach. The EAD is
calculated in a manner similar to the credit equivalent amount in Basel I
and includes the impact of netting. M is set at 2.5 in most circumstances.
Under the advanced IRB approach, banks supply their own estimates
of the PD, LGD, EAD, and M for corporate, sovereign, and bank
exposures. The PD can be reduced by credit mitigants such as credit
triggers. (As in the case of the foundation IRB approach, it is subject to a
floor of 0.03% for bank and corporate exposures.) The two main factors
influencing the LGD are the seniority of the debt and the collateral. In
calculating EAD, banks can with regulatory approval use their own
estimates of credit conversion factors.
The capital given by equation (7.8) is intended to be sufficient to cover
unexpected losses over a one-year period that we are 99% sure will not be
exceeded. Losses from the one-year "average" probability of default, PD,
should be covered by a bank in the way it prices its products. The WCDR
is the probability of default that occurs once every thousand years. The
Basel Committee reserves the right to apply a scaling factor (less than or
greater than 1.0) to the result of the calculations in equation (7.8) if it
Chapter 7
186
finds that the aggregate capital requirements are too high or low. At the
time of writing, this factor is estimated at 1.06.
Example 7.7
Suppose that the assets of a bank consist of $100 million of loans to A-rated
corporations. The PD for the corporations is estimated as 0.1% and LGD is
60%. The average maturity is 2.5 years for the corporate loans. This means
that
b = [0.11852 - 0.05478 x ln(0.00l)]2 = 0.247
so that
From Table 7.6, the WCDR is 3.4%. Under the Basel II IRB approach, the
risk-weighted assets for the corporate loans are
12.5 x 100 x 0.6 x (0.034 - 0.001) x 1.59 = 39.3
or $39.3 million. This compares with $100 million under Basel I and $50 million under the standardized approach of Basel II. (See Examples 7.1 and 7.5,
where a $100 million corporate loan is part of the portfolio.)
Retail Exposures
The model underlying the calculation of capital for retail exposures is
similar to that underlying the calculation of corporate, sovereign, and
banking exposures. However, the foundation IRB and advanced IRB
approaches are merged and all banks using the IRB approach provide
their own estimates of PD, EAD, and LGD. There is no maturity
adjustment. The capital requirement is therefore
EAD x LGD x (WCDR - PD)
and the risk-weighted assets are
12.5 x EAD x LGD x (WCDR - PD)
WCDR is calculated as in equation (7.6). For residential mortgages, is
set equal to 0.15 in this equation. For qualifying revolving exposures, is
set equal to 0.04. For all other retail exposures, a relationship between
and PD is specified for the calculation of WCDR. This is
Because exp(—35) is a very small number, this formula is to all intents
Bank Regulation and Basel II
Table 7.7
187
Relationship between WCDR and PD for
retail exposures.
PD:
0.1%
0.5%
1.0%
1.5%
2.0%
WCDR:
2.1%
6.3%
9.1%
11.0%
12.3%
and purposes
Comparing equation (7.10) with equation (7.7), we see that correlations
are assumed to be much lower for retail exposures. Table 7.7 is the table
corresponding to Table 7.6 for retail exposures.
Example 7.8
Suppose that the assets of a bank consist of $50 million of residential
mortgages where the PD is 0.005 and the LGD is 20%. In this case,
= 0.15 and
The risk-weighted assets are
12.5 x 50 x 0.2 x (0.067 - 0.005) = 7.8
or $7.8 million. This compares with $25 million under Basel I and $17.5 million under the standardized approach of Basel II. (See Examples 7.1 and 7.5,
where $50 million of residential mortgages is part of the portfolio.)
Guarantees and Credit Derivatives
The approach traditionally taken by the Basel Committee for handling
guarantees is the credit substitution approach. Suppose that an AA-rated
company guarantees a loan to a BBB-rated company. For the purposes of
calculating capital, the credit rating of the guarantor is substituted for the
credit rating of the borrower, so that capital is calculated as though the
loan had been made to the AA-rated company. This overstates the credit
risk because, for the lender to lose money, both the guarantor and the
borrower have to default (with the guarantor defaulting before the
borrower). The Basel Committee has addressed this issue. In July 2005 it
Published a document concerned with the treatment of double defaults
under Basel II. 14 As an alternative to using the credit substitution
14
See 'The Application of Basel II to Trading Activities and the Treatment of Double
Defaults," July 2005, available on www.bis.org.
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Chapter 7
approach, the capital requirement can be calculated as the capital that
would be required without the guarantee but multiplied by
0.15 + 160 x PD g , where PD g is the one-year probability of default by
the guarantor. Credit default swaps, which we talk about in Chapter 13,
provide a type of insurance against default and are handled similarly to
guarantees for regulatory purposes.
7.9 OPERATIONAL RISK UNDER BASEL II
In addition to improving the way banks calculate credit risk capital,
Basel II will require banks to keep capital for operational risk. It seems
that regulators are introducing a capital charge for operational risk for
three reasons. The first is that in an increasingly complex environment
banks face many risks arising from the possibilities of human and computer error. 15 The second is that regulators want banks to pay more
attention to their internal systems to avoid catastrophes like that at Barings
Bank. The third is that the effect of the Basel II credit risk calculation will
be to reduce the capital requirements for most banks and regulators want
another capital charge to bring the total capital back to roughly where it
was before. The regulators are currently offering three approaches:
1. The basic indicator approach
2. The standardized approach
3. The advanced measurement approach
Which of these is used depends on the sophistication of the bank. The
simplest approach is the basic indicator approach. This sets the operational risk capital equal to the bank's average annual gross income over
the last three years multiplied by 0.15.16 The standardized approach is
similar to the basic indicator approach except that a different factor is
applied to the gross income from different business lines. In the advanced
measurement approach the bank uses its own internal models to calculate
the operational risk loss that it is 99.9% certain will not be exceeded in
one year. One advantage of the advanced measurement approach is that it
15
All errors are ultimately human errors. In the case of a "computer error", someone at
a some stage made a mistake programming the computer.
16
Gross income is defined as net interest income plus noninterest income. Net interest
income is the excess of income earned on loans over interest paid on deposits and other
instruments that are used to fund the loans. Years where gross income is negative are not
included in the calculations.
Bank Regulation and Basel II
Business Snapshot 7.2
189
Basel III?
Basel II does not allow a bank to use its own credit risk diversification
calculations when setting capital requirements for credit risk within the banking book. Equation (7.6) with prescribed values of p must be used. In theory a
bank with $1 billion of lending to BBB-rated companies in a single industry is
liable to be asked to keep the same capital as a bank that has $1 billion of
lending to a much more diverse group of BBB-rated corporations. As banks
develop better credit value-at-risk models, we may see a Basel III standard
where the capital requirement in the banking book is based on a bank's own
model of the aggregate credit risks it is taking.
The total required capital under Basel II is the sum of the capital for credit
risk, market risk, and operational risk. This implicitly assumes that the risks
are perfectly correlated. For example, it assumes that the 99.9% worst-case
loss for credit risk occurs at the same time as the 99.9% worst-case loss for
operational risk. Possibly, the calculations in Basel III (if it ever comes to pass)
will allow banks to assume less than perfect correlation between losses from
different types of risk when determining regulatory capital requirements.
allows banks to recognize the risk-mitigating impact of insurance, subject
to certain conditions. We discuss the calculation of operational risk
further in Chapter 14.
There is no question that the calculations in Pillar 1 of Basel II are a
huge step forward over those in Basel I. In order to comply with Basel II,
banks are finding that they have to become much more sophisticated in
the way they handle credit risk and operational risk. Whether there will be
further major changes to the way capital requirements are calculated for
banks remains to be seen. Business Snapshot 7.2 speculates on what may
be the major changes in Basel III.
7.10 SUPERVISORY REVIEW
Pillar 2 of Basel II is concerned with the supervisory review process. Four
key principles of supervisory review are specified:
1. Banks should have a process for assessing their overall capital
adequacy in relation to their risk profile and a strategy for
maintaining their capital levels.
2. Supervisors should review and evaluate banks' internal capital
adequacy assessments and strategies, as well as their ability to
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Chapter 7
monitor and ensure compliance with regulatory capital ratios.
Supervisors should take appropriate supervisory action if they are
not satisfied with the result of this process.
3. Supervisors should expect banks to operate above the minimum
regulatory capital and should have the ability to require banks to
hold capital in excess of this minimum.
4. Supervisors should seek to intervene at an early stage to prevent
capital from falling below the minimum levels required to support
the risk characteristics of a particular bank and should require rapid
remedial action if capital is not maintained or restored.
The Basel Committee suggests that regulators pay particular attention to
interest rate risk in the banking book, credit risk, and operational risk.
Key issues in credit risk are stress tests used, default definitions used,
credit risk concentration, and the risks associated with the use of collateral, guarantees, and credit derivatives.
The Basel Committee also stresses that there should be transparency
and accountability in the procedures used by bank supervisors. This is
particularly important when a supervisor exercises discretion in the procedures used or sets capital requirements above the minimum specified in
Basel II.
7.11 MARKET DISCIPLINE
Pillar 3 of Basel II is concerned with market discipline. The Basel
Committee wants to encourage banks to increase disclosure to the market
of their risk assessment procedures and capital adequacy. The extent to
which regulators can force banks to increase disclosure varies from
jurisdiction to jurisdiction. However, banks are unlikely to ignore directives on this from their supervisors, given the potential of supervisors to
make their life difficult. Also, in some instances, banks will have to
increase their disclosure in order to be allowed to use particular methodologies for calculating capital.
Regulatory disclosures are likely to be different in form from accounting disclosures and need not be made in annual reports. It is largely left
to the bank to choose disclosures that are material and relevant. Among
the items that banks should disclose are:
1. The entities in the banking group to which Basel II is applied and
adjustments made for entities to which it is not applied
Bank Regulation and Basel II
191
2. The terms and conditions of the main features of all capital
instruments
3. A list of the instruments constituting Tier 1 capital and the amount
of capital provided by each item
4. The total amount of Tier 2 and Tier 3 capital
5. Capital requirements for credit, market, and operational risk
6. Other general information on the risks to which a bank is exposed
and the assessment methods used by the bank for different
categories of risk
7. The structure of the risk management function and how it operates
SUMMARY
This chapter has provided an overview of capital requirements for banks
throughout the world. The way in which regulators calculate the minimum capital a bank is required to hold has changed dramatically since
the 1980s. Prior to 1988, regulators determined capital requirements by
specifying minimum ratios for capital to assets or maximum ratios for
assets to capital. In the late 1980s, both bank supervisors and the banks
themselves agreed that changes were necessary. Off-balance-sheet derivatives trading was increasing fast. In addition, banks were competing
globally and it was considered important to create a level playing field
by making regulations uniform throughout the world.
The 1988 Basel Accord assigned capital for credit risk both on and off
the balance sheet. This involved calculating a risk-weighted asset for each
item. The risk-weighted asset for an On-balance-sheet loan was calculated
by multiplying the principal by a risk weight for the counterparty. In the
case of derivatives such as swaps, banks were first required to calculate a
credit equivalent amount. The risk-weighted asset was obtained by multiPlying the credit equivalent amount by a risk weight for the counterparty.
Banks were required to keep capital equal to 8% of the total risk-weighted
assets. In 1995 the capital requirements for credit risk were modified to
incorporate netting.
In 1996, the Accord was modified to include a capital charge for market
risk. Sophisticated banks could base the capital charge on a value-at-risk
calculation. In 1999, the Basel Committee proposed significant changes,
which are expected to be implemented in 2007. The capital for market risk
is unchanged. Credit risk capital will be calculated in a more sophisticated
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Chapter 7
way that will reflect either (a) credit ratings from agencies such as Moody's
or S&P or (b) a bank's own internal estimates of default probabilities. In
addition, there will be a capital requirement for operational risk.
FURTHER READING
Bank for International Settlements, "Basel II: International Convergence of
Capital Measurement and Capital Standards: A Revised Framework,"
November 2005, www.bis.org.
Crouhy, M., D. Galai, and R. Mark, Risk Management. New York: McGrawHill, 2001.
Gordy, M.B., "A Risk-factor Model Foundation for Ratings-Based Bank
Capital Ratios," Journal of Financial Intermediation, 12 (2003): 199-232.
Lopez, J. A., "The Empirical Relationship Between Average Asset Correlation,
Firm Probability of Default, and Asset Size," Journal of Financial
Intermediation, 13, No. 2 (2004): 265-283.
Vasicek, O., "Probability of Loss on a Loan Portfolio," Working Paper, KMV,
1987. [Published in Risk in December 2002 under the title "Loan Portfolio
Value".]
QUESTIONS AND PROBLEMS (Answers at End of Book)
7.1. "When a steel company goes bankrupt, other companies in the same
industry benefit because they have one less competitor. But when a bank
goes bankrupt, other banks do not necessarily benefit." Explain this
statement.
7.2. "The existence of deposit insurance makes it particularly important for
there to be regulations on the amount of capital banks hold." Explain this
statement.
7.3. As explained in Section 2.3 an interest rate swap involves the exchange of a
fixed rate of interest for a floating rate of interest with both being applied to
the same principal. The principals are not exchanged. What is the nature of
the credit risk for a bank when it enters into a five-year interest rate swap
with a notional principal of $100 million? Assume the swap is worth zero
initially.
7.4. In a currency swap, interest on a principal in one currency is exchanged for
interest on a principal in another currency. The principals in the two
currencies are exchanged at the end of the life of the swap. Why is the
credit risk on a currency swap greater than that on an interest rate swap?
Bank Regulation and Basel II
193
7.5. An interest rate swap currently has a negative value to a financial
institution. Is the financial institution exposed to credit risk on the transaction? Explain your answer.
7.6. Estimate the capital required under Basel I for a bank that has the following
transactions with a corporation (assume no netting): (a) a 9-year interest
rate swap with a notional principal of $250 million and a current market
value of — $2 million; (b) a 4-year interest rate swap with a notional
principal of $100 million and a current value of $3.5 million; and (c) a
6-month derivative on a commodity with a principal of $50 million that is
currently worth $1 million.
7.7. What is the capital required in Problem 7.6 under Basel I assuming that
the 1995 netting amendment applies?
7.8. All the contracts a bank has with a corporate client are loans to the client.
What is the value to the bank of netting provisions in the loan agreement?
7.9. Explain why the final stage in the Basel II calculations for credit risk,
market risk, and operational risk is to multiply by 12.5.
7.10. What is the difference between the trading book and the banking book for
a bank? A bank currently has a loan of $10 million dollars to a corporate
client. At the end of the life of the loan the client would like to sell debt
securities to the bank instead of borrowing. How does this change affect
the nature of the bank's regulatory capital calculations?
7.11. Under Basel I, banks do not like lending to highly creditworthy companies
and prefer to help them issue debt securities. Why is this? Do you expect
the banks' attitude to this type of lending to change under Basel II?
7.12. What is regulatory arbitrage?
7.13. Equation (7.8) gives the formula for the capital required under Basel II.
It involves four terms being multiplied together. Explain each of these
terms.
7.14. Explain the difference between the simple and the comprehensive
approach for adjusting for collateral.
7.15. Explain the difference between the standardized approach, the IRB
approach, and the advanced IRB approach for calculating credit risk
capital under Basel II.
7.16. Explain the difference between the basic indicator approach, the standardized approach, and the advanced measurement approach for calculating operational risk capital under Basel II.
7.17. Suppose that the assets of a bank consist of $200 of retail loans (not
mortgages). The PD is 1% and the LGD is 70%. What is the riskweighted assets under the Basel II IRB approach? How much Tier 1
and Tier 2 capital is required.
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Chapter 7
ASSIGNMENT QUESTIONS
7.18. Why is there an add-on amount in Basel I for derivatives transactions?
"Basel I could be improved if the add-on amount for a derivatives transaction depended on the value of the transaction." How would you argue
this viewpoint?
7.19. Estimate the capital required under Basel I for a bank that has the following
transactions with another bank (assume no netting): (a) a 2-year forward
contract on a foreign currency, currently worth $2 million, to buy foreign
currency worth $50 million; (b) a long position in a 6-month option on the
S&P 500 with a principal of $20 million and a current value of $4 million;
and (c) a 2-year swap involving oil with a principal of $30 million and a
current swap value of -$5 million. What difference does it make if the
netting amendment applies?
7.20. A bank has the following transaction with an AA-rated corporation:
(a) a 2-year interest rate swap with a principal of $100 million worth
$3 million; (b) a 9-month foreign exchange forward contract with a
principal of $150 million worth -$5 million; and (c) a 6-month long
option on gold with a principal of $50 worth $7 million. What is the
capital requirement under Basel I if there is no netting? What difference
does it make if the netting amendment applies? What is the capital
required under Basel II when the standardized approach is used?
7.21. Suppose that the assets of a bank consist of $500 million of loans to
BBB-rated corporations. The PD for the corporations is estimated as
0.3%. The average maturity is 3 years and the LGD is 60%. What is the
risk-weighted assets for credit risk under the Basel II advanced IRB
approach? How much Tier 1 and Tier 2 capital is required. How does this
compare with the capital required under the Basel II standardized approach
and under Basel I?
The VaR Measure
In Chapter 3 we examined measures such as delta, gamma, and vega for
describing different aspects of the risk in a portfolio of derivatives. A
financial institution usually calculates each of these measures each day for
every market variable to which it is exposed. Often there are hundreds, or
even thousands, of these market variables. A delta-gamma-vega analysis
therefore leads to a huge number of different risk measures being
produced each day. These risk measures provide valuable information
for a trader who is responsible for managing the part of the financial
institution's portfolio that is dependent on a particular market variable.
However, they do not provide a way of measuring the total risk to which
the financial institution is exposed.
Value at risk (VaR) is an attempt to provide a single number that
summarizes the total risk in a portfolio of financial assets. It was
pioneered by J.P. Morgan (see Business Snapshot 8.1), and it has become
widely used by corporate treasurers and fund managers as well as by
financial institutions. As we saw in Chapter 7, the VaR measure is used
by the Basel Committee in setting capital requirements for banks
throughout the world.
In this chapter we explain the VaR measure and discuss its strengths
and weaknesses. We also cover back testing and stress testing. In the next
two chapters we will explain the two main approaches for estimating VaR
for market risk. In Chapter 12 we will consider how VaR can be estimated
for credit risk.
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Business Snapshot 8.1
Historical Perspectives on VaR
J.P. Morgan is credited with helping to make VaR a widely accepted measure.
The Chairman, Dennis Weatherstone, was dissatisfied with the long risk
reports he received every day. These contained a huge amount of detail on
the Greek letters for different exposures, but very little that was really useful to
top management. He asked for something simpler that focused on the bank's
total exposure over the next 24 hours measured across the bank's entire trading
portfolio. At first his subordinates said this was impossible, but eventually
they adapted the Markowitz portfolio theory (see Section 1.1) to develop a
VaR report. This became known as the 4:15 report because it was placed on
the chairman's desk at 4:15 p.m. every day after the close of trading.
Producing the report entailed a huge amount of work involving the collection of data daily on the positions held by the bank around the world, the
handling of different time zones, the estimation of correlations and volatilities,
and the development of computer systems. The work was completed in about
1990. The main benefit of the new system was that senior management had a
better understanding of the risks being taken by the bank and were better able
to allocate capital within the bank. Other banks had been working on similar
approaches for aggregating risks and by 1993 VaR was established as an
important risk measure.
Banks usually keep the details about the models they develop internally a
secret. However, in 1994 J.P. Morgan made a simplified version of their own
system, which they called RiskMetrics, available on the internet. RiskMetrics
included variances and covariances for a very large number of different market
variables. This attracted a lot of attention and led to debates about the pros and
cons of different VaR models. Software firms started offering their own VaR
models, some of which used the RiskMetrics database. After that, VaR was
rapidly adopted as a standard by financial institutions and some nonfinancial
institutions. The BIS Amendment, which was based on VaR (see Section 7.6),
was announced in 1996 and implemented in 1998. Later the RiskMetrics group
within J.P. Morgan was spun off as a separate company. This company
developed CreditMetrics for handling credit risks in 1997 and CorporateMetrics for handling the risks faced by nonfinancial corporations in 1999.
8.1 DEFINITION OF VaR
When using the value-at-risk measure, we are interested in making a
statement of the following form:
We are X percent certain that we will not lose more than V dollars in the
next N days.
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197
The variable V is the VaR of the portfolio. It is a function of two
parameters: the time horizon (N days) and the confidence level (X%). It
is the loss level over N days that we are X% certain will not be exceeded.
VaR is the loss corresponding to the (100 — X)th percentile of the
distribution of the change in the value of the portfolio over the next
N days. (Gains are positive changes; losses are negative changes.) For
example, when N = 5 and X = 97, VaR is the third percentile of the
distribution of changes in the value of the portfolio over the next five
days. In Figure 8.1, VaR is illustrated for the situation where the change
in the value of the portfolio is approximately normally distributed.
Figure 8.1 shows the distribution of the portfolio's daily gain, with
losses being counted as negative gains. As mentioned, VaR is the
(100 — X)th percentile of this distribution. Instead, we can calculate the
distribution of the portfolio's daily loss, with a gain being counted as a
negative loss. VaR is the Xth percentile of this distribution.
As discussed in Section 7.6, the 1996 BIS Amendment calculates
capital for the trading book using the VaR measure with N = 10 and
X = 99. This means that it focuses on the revaluation loss over a 10-day
period that is expected to be exceeded only 1 % of the time. The capital it
requires the bank to hold is k times this VaR measure (with an adjustment
for what are termed specific risks.) The multiplier k is chosen on a bankby-bank basis by the regulators and must be at least 3.0. For a bank with
excellent well-tested VaR estimation procedures, it is likely that k will be
set equal to the minimum value of 3.0. For other banks it may be higher.
As we will discuss in Section 8.6, when tests show a bank's VaR model
would not have performed well during the last 250 days, k may be as high
as 4.0.
Figure 8.1
Calculation of VaR from the probability distribution of the change
in the portfolio value; confidence level is X%.
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8.2 VaR vs. EXPECTED SHORTFALL
VaR is an attractive measure because it is easy to understand. In essence,
it asks the simple question: "How bad can things get?" This is the
question all senior managers want answered. They are very comfortable
with the idea of compressing all the Greek letters for all the market
variables underlying a portfolio into a single number. VaR is also
relatively easy to back test, as we shall see later in this chapter.
However, when VaR is used in an attempt to limit the risks taken by a
trader, it can lead to undesirable results. Suppose that a bank tells a
trader that the one-day 99% VaR of the trader's portfolio must be kept at
less than $10 million. The trader can construct a portfolio where there is
a 99% chance that the daily loss is less than $10 million and a 1% chance
that it is $500 million. The trader is satisfying the risk limits imposed by
the bank but is clearly taking unacceptable risks.
This behavior by a trader is not as unlikely as it sounds. Many traders
like taking high risks in the hope of realizing high returns. If they can find
ways of taking high risks without violating risk limits, they will do so. To
quote one trader the author has talked to: "I have never met a risk control
system that I cannot trade around." The problem we are talking about is
summarized by Figures 8.1 and 8.2. The figures show the probability
distribution for the gain or loss on a portfolio during N days. Both
portfolios have the same VaR, but the portfolio in Figure 8.2 is much
riskier than that in Figure 8.1 because expected losses are much larger.
Expected Shortfall
A measure that produces better incentives for traders than VaR is expected
shortfall. This is also sometimes referred to as conditional VaR or tail loss.
Figure 8.2
Alternative situation to Figure 8.1; VaR is the same, but the
potential loss is larger.
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199
Whereas VaR asks how bad can things get, expected shortfall asks: "If
things do get bad, what is the expected loss?" Expected shortfall, like VaR,
is a function of two parameters: N (the time horizon in days) and X (the
percent confidence level). It is the expected loss during an N-day period
conditional on the loss being greater than the Xth percentile of the loss
distribution.1 For example, with X = 99 and N = 10, the expected shortfall is the average amount we lose over a ten-day period assuming that the
loss is greater than the 99th percentile of the loss distribution.
As we show in the next section, expected shortfall has better properties
than VaR in that it encourages diversification. One disadvantage is that it
does not have the simplicity of VaR and, as a result, is slightly more
difficult to understand. Another is that it is more difficult to back test.
VaR has become the most popular measure of risk among both regulators
and risk managers in spite of its weaknesses. Therefore, in most of our
discussions in this chapter and the next two, we will focus on how VaR can
be measured and used. Many of the points we make apply equally to
expected shortfall and other risk measures.
8.3 PROPERTIES OF RISK MEASURES
A risk measure used for specifying capital requirements can be thought of
as the amount of cash (or capital) that must be added to a position to make
its risk acceptable to regulators. Artzner et al. have proposed a number of
properties that such a risk measure should have.2 These are:
1. Monotonicity: If a portfolio has lower returns than another portfolio
for every state of the world, its risk measure should be greater.
2. Translation invariance: If we add an amount of cash K to a portfolio,
its risk measure should go down by K.
3. Homogeneity: Changing the size of a portfolio by a factor while
keeping the relative amounts of different items in the portfolio the same
should result in the risk measure being multiplied by
4. Subadditivity: The risk measure for two portfolios after they have been
merged should be no greater than the sum of their risk measures before
they were merged.
1
As mentioned earlier, gains are calculated as negative losses, and so all outcomes are
considered when a loss distribution is constructed.
See P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath, "Coherent Measures of Risk,"
Mathematical Finance, 9 (1999): 203-228.
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Chapter 8
The first three conditions are straightforward, given that the risk measure
is the amount of cash needed to be added to the portfolio to make its risk
acceptable. The fourth condition states that diversification helps reduce
risks. When we aggregate two risks, the total of the risk measures
corresponding to the risks should either decrease or stay the same. VaR
satisfies the first three conditions. However, it does not always satisfy the
fourth one, as is illustrated by the following example.
Example 8.1
Consider two $10 million one-year loans each of which has a 1.25% chance of
defaulting. If a default occurs on one of the loans, the recovery of the loan
principal is uncertain, with all recoveries between 0% and 100% being equally
likely. If the loan does not default, a profit of $0.2 million is made. To simplify
matters, we suppose that if one loan defaults then it is certain that the other
loan will not default.3
For a single loan, the one-year 99% VaR is $2 million. This is because there
is a 1.25% chance of a loss occurring, and conditional on a loss there is an
80% chance that the loss is greater than $2 million. The unconditional
probability that the loss is greater than $2 million is therefore 80% of
1.25%, or 1%.
Consider next the portfolio of two loans. Each loan defaults 1.25% of the
time and they never default together. There is therefore a 2.5% probability
that a default will occur. The VaR in this case turns out to be $5.8 million.
This is because there is a 2.5% chance of one of the loans defaulting, and
conditional on this event there is an 40% chance that the loss on the loan that
defaults is greater than $6 million. The unconditional probability that the loss
on the defaulting loan is greater than $6 million is therefore 40% of 2.5%, or
1%. A profit of $0.2 million is made on the other loan showing that the oneyear 99%VaR is $5.8 million.
The total VaR of the loans considered separately is 2 + 2 = 4 million. The
total VaR after they have been combined in the portfolio is $1.8 million
greater at $5.8 million. This is in spite of the fact that there are very attractive
diversification benefits from combining the loans in a single portfolio.
Coherent Risk Measures
Risk measures satisfying all four conditions are referred to as coherentExample 8.1 illustrates that VaR is not coherent. It can be shown that the
expected shortfall measure we discussed earlier is coherent. The following
3
This is to simplify the calculations. If the loans default independently of each other, so
that two defaults can occur, the numbers are slightly different and the VaR of the
portfolio is still greater than the sum of the VaRs of the individual loans.
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201
example illustrates this by calculating expected shortfalls for the situation
in Example 8.1.
Example 8.2
Consider again the situation in Example 8.1. We showed that the VaR for a
single loan is $2 million. The expected shortfall from a single loan when the
time horizon is one year and the confidence level is 99% is therefore the
expected loss on the loan conditional on a loss greater than $2 million. Given
that losses are uniformly distributed between zero and $10 million, this is
halfway between $2 million and $10 million, or $6 million.
The VaR for a portfolio consisting of the two loans was calculated in
Example 8.1 as $5.8 million. The expected shortfall from the portfolio is
therefore the expected loss on the portfolio conditional on the loss being
greater than $5.8 million. When a loan defaults, the other (by assumption)
does not and outcomes are uniformly distributed between a gain of $0.2 million and a loss of $9.8 million. The expected loss given that we are in the part
of the distribution between $5.8 million and $9.8 million is $7.8 million. This
is therefore the expected shortfall of the portfolio.
Because 6 + 6 > 7.8, the expected shortfall does satisfy the Subadditivity
condition.
A risk measure can be characterized by the weights it assigns to quantiles
of the loss distribution.4 VaR gives a 100% weighting to the Xth quantile
and zero to other quantiles. Expected shortfall gives equal weight to all
quantiles greater than the Xth quantile and zero weight to all quantiles
below the Xth quantile. We can define what is known as a spectral risk
measure by making other assumptions about the weights assigned to
quantiles. A general result is that a spectral risk measure is coherent
(i.e., it satisfies the Subadditivity condition) if the weight assigned to the
qth quantile of the loss distribution is a nondecreasing function of q.
Expected shortfall satisfies this condition. However, VaR does not,
because the weights assigned to quantiles greater than X are less than
the weight assigned to the Xth quantile. Some researchers have proposed
measures where the weights assigned to the qth quantile of the loss
distribution increase relatively fast with q. One idea is to make the weight
assigned to the qth quantile proportional to
where
is a
constant. This is referred to as the exponential spectral risk measure.
Figure 8.3 shows the weights assigned to loss quantiles for expected
shortfall and for the exponential spectral risk measure when has two
different values.
4
Quantiles are also referred to as percentiles or fractiles.
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Chapter 8
Figure 8.3 Weights as a function of quantiles for (a) expected shortfall
when X — 90%, (b) exponential spectral risk measure with =0.15, and
(c) exponential spectral risk measure with = 0.05.
8.4 CHOICE OF PARAMETERS FOR VaR
We now return to a consideration of VaR. The user must choose two
parameters: the time horizon and the confidence level. As mentioned in
Chapter 7, the Basel Committee has chosen a time horizon of ten days
and a confidence level of 99% for market risks in the trading book. It has
also chosen a time horizon of one year and a confidence level of 99.9%
for credit risks under the internal-ratings-based approach and for operational risk under the advanced measurement approach. Other parameter
values are chosen in different situations. For example, Microsoft in its
financial statements says that it calculates VaR with a 97.5% confidence
level and a 20-day time horizon.
A common, though questionable, assumption is that the change in the
portfolio value over the time horizon is normally distributed. The mean
change in the portfolio value is usually assumed to be zero. These assumptions are convenient because they lead to a simple formula for VaR:
(8.1)
where X is the confidence level, is the standard deviation of the portfolio change over the time horizon, and
is the inverse cumulative
normal distribution (which can be calculated using NORMSINV in
Excel). This equation shows that, regardless of the time horizon, VaR
for a particular confidence level is proportional to
The VaR Measure
203
Example 8.3
Suppose that the change in the value of a portfolio over a ten-day time horizon
is normal with a mean of zero and a standard deviation of $20 million. The tenday 99% VaR is
or $46.5 million.
The Time Horizon
An appropriate choice for the time horizon depends on the application.
The trading desks of banks calculate the profit and loss daily. Their
positions are usually fairly liquid and actively managed. For internal
use, it therefore makes sense to calculate a VaR over a time horizon of
one trading day. If VaR is unacceptable, the portfolio can be adjusted
fairly quickly. Also, a VaR with a longer time horizon might not be
meaningful because of changes in the composition of the portfolio.
For an investment portfolio held by a pension fund, a time horizon of
one month is often chosen. This is because the portfolio is traded less
actively and some of the instruments in the portfolio are less liquid. Also
the performance of pension fund portfolios is often monitored monthly.
Whatever the application, when market risks are being considered
analysts almost invariably start by calculating VaR for a time horizon
of one day. The usual assumption is
This formula is exactly true when the changes in the value of the portfolio
on successive days have independent identical normal distributions with
mean zero. In other cases it is an approximation. The formula follows
from equation (8.1) and the following results:
1. The standard deviation of the sum on N independent identically
distributions is
times the standard deviation of each distribution
2. The sum of independent normal distributions is normal
Regulatory Capital
As mentioned earlier, the regulatory capital for market risk in the trading
book is based on the ten-day 99% VaR. Regulators explicitly state that
the ten-day 99% VaR can be calculated using equation (8.2) as
times
the one-day 99% VaR. This means that when the capital requirement for
a bank is specified as three times the ten-day 99 % VaR it is to all intents
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Chapter 8
and purposes 3 x
=9.49 times the one-day 99% VaR. In the next
two chapters we will focus entirely on the calculation of one-day VaR for
market risks.
Impact of Autocorrelation
In practice, the changes in the value of a portfolio from one day to the next
are not always totally independent. Define
as the change in the value
of a portfolio on day i. A simple assumption is first-order autocorrelation
where the correlation between
and
is for all i. Suppose that the
variance of
is
for all i. Using the usual formula for the variance of
the sum of two variables, the variance of
is
The correlation between
formula for the variance of
and
is
This
leads to the following
(see Problem 8.12):
(8.3)
Table 8.1 shows the impact of autocorrelation on the N-day VaR that is
calculated from the one-day VaR. It assumes that the distribution of daily
changes in the portfolio are identical normals with mean zero. Note that
the ratio of the N-day VaR to the one-day VaR does not depend on the
daily standard deviation or on the confidence level. This follows from
the result in equation (8.1) and the property of equation (8.3) that the
N-day standard deviation is proportional to the one-day standard deviation. Comparing the = 0 row in Table 8.1 with the other rows shows that
the existence of autocorrelation results in the VaR estimates calculated
from equation (8.1) being a little low.
Table 8.1 Ratio of TV-day VaR to one-day VaR for different values of N when
there is first-order correlation; distribution of change in portfolio value each day
is assumed to have the same normal distribution with mean zero; is the
autocorrelation parameter.
= 0
= 0.05
= 0.1
= 0.2
N= 1
N = 2
1.00
1.00
1.00
1.00
1.41
1.45
1.48
1.55
N=5
2.24
2.33
2.42
2.62
N = 10
N = 50
N = 250
3.16
3.31
3.46
3.79
7.07
7.43
7.80
8.62
15.81
16.62
17.47
19.35
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205
Example 8.4
Suppose that the standard deviation of daily changes in the portfolio value is
$3 million and the first-order autocorrelation of daily changes is 0.1. From
equation (8.3), the variance of the change in the portfolio value over five days is
32[5 + 2 x 4 x 0 . 1 + 2 x 3 x 0.12 + 2 x 2 x 0.13 + 2 x 1 x 0.14] = 52.7778
The standard deviation of the change in the value of the portfolio over five days
is
or 7.265. The five-day 95% VaR is therefore
or $11.95 million. Note that the ratio of the fiveday standard deviation of portfolio changes to the one-day standard deviation
is 7.265/3 = 2.42. Since VaRs are proportional to standard deviations under
the assumptions we are making, this is the number in Table 8.1 for = 0.1 and
N = 5.
Confidence Level
The confidence level chosen for VaR is likely to depend on a number of
factors. Suppose a bank wants to maintain an AA credit rating and
calculates that companies with this credit rating have a 0.03% chance
of defaulting over a one-year period. It might choose to use a 99.97%
confidence level in conjunction with a one-year time horizon for internal
risk management purposes. (It might also communicate the analysis to
rating agencies as evidence that it deserves its AA rating.)
The confidence level that is actually used for the first VaR calculation is
often much less than the one that is eventually reported. This is because it
is very difficult to estimate a VaR directly when the confidence level is
very high. If daily portfolio changes are assumed to be normally distributed with zero mean, we can use equation (8.1) to convert a VaR
calculated with one confidence level to a VaR with another confidence
level. Suppose that
is the standard deviation of the change in the
portfolio value over a certain time horizon and that the expected change
in the portfolio value is zero. Denote VaR for a confidence level of X by
VaR(X). From equation (8.1), we have
for all confidence levels X. It follows that
Unfortunately this formula is critically dependent on the shape of the
tails of the loss distribution being normal. When they are not normal, the
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Chapter 8
formula may be quite a bad approximation. Extreme value theory, which
is covered in Chapter 9, provides an alternative way of extrapolating tails
of loss distributions.
Equation (8.4) assumes that the two VaR measures have the same time
horizon. If we want to change the time horizon, we can use equation (8.4)
in conjunction with equation (8.2) or (8.3).
Example 8.5
Suppose that the one-day VaR with a confidence level of 95% is $1.5 million.
Using the assumption that the distribution of portfolio value changes is
normal with mean zero, the one-day 99% VaR is
or $2.12 million. If we assume daily changes are independent, the ten-day
99% VaR is
times this or $6.71 million and the 250-day VaR is
times this, or $33.54 million.
8.5 MARGINAL VaR, INCREMENTAL VaR, AND
COMPONENT VaR
Analysts often calculate additional measures in order to understand VaR.
Consider a portfolio with a number of components where the investment
in the ith component is
The marginal VaR is the sensitivity of VaR to
the amount invested in the ith component. It is
For an investment portfolio, marginal VaR is closely related to the capital
asset pricing model's beta (see Section 1.1). If an asset's beta is high, its
marginal VaR will tend to be high; if its beta is low, the marginal VaR
tends to be low. In some circumstances marginal VaR is negative,
indicating that increasing the weighting of a particular asset reduces the
risk of the portfolio.
Incremental VaR is the incremental effect on VaR of a new trade or the
incremental effect of closing out an existing trade. It asks the question:
"What is the difference between VaR with and without the trade." It is of
particular interest to traders who are wondering what the effect of a new
trade will be on their regulatory capital. If a component is small in
relation to the size of a portfolio, it may be reasonable to assume that
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207
the marginal VaR remains constant as
is reduced all the way to zero.
This leads to the following approximate formula for the incremental VaR
of the ith component:
The component VaR for the ith component of the portfolio is the part
of the VaR of the portfolio that can be attributed to this component.
Component VaRs should have the following properties:
1. The ith component VaR for a large portfolio should be approximately equal to the incremental VaR for that component.
2. The sum of all the component VaRs should equal the portfolio
VaR.
Owing to nonlinearities in the calculation of VaR, we cannot satisfy the
first condition exactly if we also want to satisfy the second condition. A
result known as Euler's theorem can be used to calculate component
VaRs. This is5
where N is the number of components. We can therefore set
where Ci is the component VaR for the ith component. From the Euler's
theorem result, these satisfy the second condition specified above:
Also, as indicated by equation (8.5), they satisfy the first condition.
Interestingly, the definition of Ci in equation (8.6) is equivalent to the
alternative definition that Ci is the expected loss on the ith position,
conditional on the loss on the portfolio equaling the VaR level.
Marginal, incremental, and component expected shortfall can be defined analogously to marginal, incremental, and component VaR. Euler's
The condition that we need for Euler's theorem is that when each of the xi is multiplied
by A. the portfolio VaR is multiplied by This condition is known as linear homogeneity
and is clearly satisfied.
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Chapter 8
theorem applies, so component expected shortfall can be defined by
equation (8.6) with VaR replaced by expected shortfall.
8.6 BACK TESTING
Whatever the method used for calculating VaR, an important reality
check is back testing. It involves testing how well the VaR estimates
would have performed in the past. Suppose that we have developed a
procedure for calculating a one-day 99% VaR. Back testing involves
looking at how often the loss in a day exceeded the one-day 99% VaR
calculated using the procedure for that day. Days when the actual change
exceeds VaR are referred to as exceptions. If exceptions happen on about
1 % of the days, we can feel reasonably comfortable with the methodology for calculating VaR. If they happen on, say, 7% of days, the
methodology is suspect and it is likely that VaR is underestimated. From
a regulatory perspective, the capital calculated using the VaR estimation
procedure is then too low. On the other hand, if exceptions happen on,
say 0.3% of days it is likely that the procedure is overestimating VaR and
the capital calculated is too high.
One issue in back testing VaR is whether we take account of changes
made in the portfolio during the time period considered. There are two
possibilities. The first is to compare VaR with the hypothetical change in
the portfolio value calculated on the assumption that the composition of
the portfolio remains unchanged during the time period. The other is to
compare VaR to the actual change in the value of the portfolio during the
time period. VaR itself is invariably calculated on the assumption that the
portfolio will remain unchanged during the time period, and so the first
comparison based on hypothetical changes is more logical. However, it is
actual changes in the portfolio value that we are ultimately interested in.
In practice, risk managers usually compare VaR to both hypothetical
portfolio changes and actual portfolio changes. (In fact, regulators insist
on seeing the results of back testing using actual as well as hypothetical
changes.) The actual changes are adjusted for items unrelated to the
market risk—such as fee income and profits from trades carried out at
prices different from the mid-market price.
Suppose that the time horizon is one day and the confidence limit is
X%. If the VaR model used is accurate, the probability of the VaR
being exceeded on any given day is p = 1 - X. Suppose that we look at
a total of n days and we observe that the VaR limit is exceeded on m of
The VaR Measure
209
the days where m/n > p. Should we reject the model for producing
values of VaR that are too low? Expressed formally, we can consider
two alternative hypotheses:
1. The probability of an exception on any given day is p.
2. The probability of an exception on any given day is greater than p.
From the properties of the binomial distribution, the probability of the
VaR limit being exceeded on m or more days is
This can be calculated using the BINOMDIST function in Excel. An
often-used confidence level in statistical tests is 5%. If the probability of
the VaR limit being exceeded on m or more days is less than 5%, we reject
the first hypothesis that the probability of an exception is p. If this
probability of the VaR limit being exceeded on m or more days is greater
than 5%, then the hypothesis is not rejected.
Example 8.6
Suppose that we back test a VaR model using 600 days of data. The VaR
confidence level is 99% and we observe nine exceptions. The expected number
of exceptions is six. Should we reject the model? The probability of nine or
more exceptions can be calculated in Excel as
1 - BINOMDIST(8, 600, 0.01, TRUE)
It is 0.152. At a 5% confidence level, we should not therefore reject the model.
However, if the number of exceptions had been 12, we would have calculated
the probability of 12 or more exceptions as 0.019 and rejected the model. The
model is rejected when the number of exceptions is 11 or more. (The probability of 10 or more exceptions is greater than 5%, but the probability of 11
or more is less than 5%.)
When the number of exceptions, m, is lower than the expected number of
exceptions, we can similarly test whether the true probability of an
exception is 1%. (In this case, our alternative hypothesis is that the true
probability of an exception is less than 1%.) The probability of m or less
exceptions is
and this is compared with the 5% threshold.
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Chapter 8
Example 8.7
Suppose again that we back test a VaR model using 600 days of data when the
VaR confidence level is 99% and we observe one exception, well below the
expected number of six. Should we reject the model? The probability of one or
zero exceptions can be calculated in Excel as
BINOMDIST(l, 600, 0.01, TRUE)
It is 0.017. At a 5% confidence level, we should therefore reject the model.
However, if the number of exceptions had been two or more, we would not
have rejected the model.
The tests we have considered so far have been one-tailed tests. In
Example 8.6 we assumed that the true probability of an exception was
either 1% or greater than 1%. In Example 8.7 we assumed that it was 1%
or less than 1 %. Kupiec has proposed a relatively powerful two-tailed test.6
If the probability of an exception under the VaR model is p and m
exceptions are observed in n trials, then
should have a chi-square distribution with one degree of freedom. Values
of the statistic are high for either very low or very high numbers of
exceptions. There is a probability of 5% that the value of a chi-square
variable with one degree of freedom will be greater than 3.84. It follows
that we should reject the model whenever the expression in equation (8.7)
is greater than 3.84.
Example 8.8
Suppose that as in the previous two examples we back test a VaR model using
600 days of data when the VaR confidence level is 99%. The value of the
statistic in equation (8.7) is greater that 3.84 when the number of exceptions, m,
is one or less and when the number of exceptions is 12 or more. We therefore
accept the VaR model when 2 m 11, and reject it otherwise.
Generally speaking the difficulty of back testing a VaR model increases as
the VaR confidence level increases. This is an argument in favor of not
using very high confidence levels for VaR.
Basel Committee Rules
The 1986 BIS Amendment (see Section 7.6) requires VaR models to be
back tested. Banks should use both actual and hypothetical changes in
6
See P. Kupiec, "Techniques for Verifying the Accuracy of Risk Management Models,
Journal of Derivatives, 3 (1995), 73-84.
The VaR Measure
211
the daily profit and loss to test a one-day VaR model that has a
confidence level of 99%. If the number of exceptions during the previous
250 days is less than 5, the regulatory multiplier for VaR is set at its
minimum value of 3. If the number of exceptions are 5, 6, 7, 8, and 9,
values of the multiplier equal to 3.4, 3.5, 3.65, 3.75, and 3.85, respectively,
are specified. The bank supervisor has some discretion as to whether the
higher multipliers are used. The penalty will normally apply when the
reason for the exceptions is identified as a deficiency in the VaR model
being used. If changes in positions during the day result in exceptions, the
higher multiplier should be considered. When the only reason that is
identified is bad luck, no guidance is provided for the supervisor. In
circumstances where the number of exceptions is 10 or more the Basel
Amendment requires the multiplier to be set at 4. The statistical tests we
have presented can be used to determine the confidence limits the Basel
Committee is implicitly using for its decision to accept or reject a model
(see Problem 8.13).
Bunching
A separate issue from the number of exceptions is bunching. If daily
portfolio changes are independent, exceptions should be spread evenly
through the period used for back testing. In practice, they are often
bunched together, suggesting that losses on successive days are not
independent. One approach for testing for bunching is to use the test
for autocorrelation in Section 5.9. Another approach is to use the
following test statistic suggested by Christofferson7
where
is the number of observations in which we go from a day where
we are in state i to a day where we are in state j. This statistic is chi-square
with one degree of freedom if there is no bunching. State 0 is a day where
there is no exception while state 1 is a day where there is an exception.
Also,
7
See P. F. Christofferson, "Evaluating Interval Forecasts," International Economic
Review, 39 (1998), 841-862.
212
Chapter 8
8.7 STRESS TESTING
In addition to requiring that a model for market risk be back tested, the
Basel Committee requires market risk VaR calculations be accompanied
by a "rigorous and comprehensive" stress-testing program. Stress testing
involves estimating how the portfolio would have performed under extreme
market moves. These extreme market moves typically have a very low
(virtually zero) probability under most VaR models—but they do happen!
Stress testing is a way of taking into account extreme events that are
virtually impossible according to the probability distributions assumed
for market variables, but do occur from time to time. A five-standarddeviation daily move in a market variable is one such extreme event.
Under the assumption of a normal distribution, it happens about once
every 7,000 years, but, in practice, it is not uncommon to see a fivestandard-deviation daily move once or twice every ten years.
Some stress tests focus on particular market variables. Examples of
stress tests that have been recommended include:
1.
2.
3.
4.
Shifting a yield curve by 100 basis points
Changing implied volatilities for an asset by 20% of current values
Changing an equity index by 10%
Changing an exchange rate for a major currency by 6% or changing
the exchange rate for a minor currency by 20%
Stress tests more often involve making changes to several market variables. A common practice is to use historical scenarios. For example, to
test the impact of an extreme movement in US equity prices, a company
might set the percentage changes in all market variables equal to those on
October 19, 1987 (when the S&P 500 moved by 22.3 standard deviations).
If this is considered too extreme, the company might choose January 8,
1988 (when the S&P 500 moved by 6.8 standard deviations). To test the
effect of extreme movements in UK interest rates, the company might set
the percentage changes in all market variables equal to those on April 10,
1992 (when ten-year bond yields moved by 8.7 standard deviations).
The scenarios used in stress testing are also sometimes generated by
senior management. One technique sometimes used is to ask senior
management to meet periodically and "brainstorm" to develop extreme
scenarios that might occur given the current economic environment and
global uncertainties. Whatever the procedure used to generate the stress
tests, there should be a "buy in" to the idea of stress testing by senior
The VaR Measure
213
management and it should be senior management that reviews the results
of stress tests.
If movements in only a few variables are specified in a stress test, one
approach is to set changes in all other variables to zero. Another
approach is to regress the nonstressed variables on the variables that
are being stressed to obtain forecasts for them, conditional on the
changes being made to the stressed variables. These forecasts can be
incorporated into the stress test. This is known as conditional stress testing
and is discussed by Kupiec.8
SUMMARY
A value-at-risk (VaR) calculation is aimed at making a statement of the
form: "We are X percent certain that we will not lose more than V dollars
in the next N days." The variable V is the VaR, X% is the confidence
level, and N days is the time horizon. It has become a very popular risk
measure. An alternative measure that has rather better theoretical properties is expected shortfall. This is the expected loss conditional on the loss
being greater than the VaR level.
When changes in a portfolio value are normally distributed, a VaR
estimate with one confidence level can be used to calculate a VaR level
with another confidence level. Also, if one-day changes have independent
normal distributions, an N-day VaR equals the one-day VaR multiplied
by
When the independence assumption is relaxed other somewhat
more complicated formulas can be used to go from the one-day VaR to
the N-day VaR.
The marginal VaR with respect to the ith position is the partial
derivative of VaR with respect to the size of the position. The incremental
VaR with respect to a particular position is the incremental effect of that
position on VaR. There is a formula that can be used for dividing VaR
into components that correspond to the positions taken. The component
VaRs sum to VaR and each component is, for a large portfolio of
relatively small positions, approximately equal to the corresponding
incremental VaR.
Back testing is an important part of a VaR system. It examines how
well the VaR model would have performed in the past. There are two
ways in which back testing may indicate weaknesses in a VaR model. One
8
P. Kupiec, "Stress Testing in a Value at Risk Framework," Journal of Derivatives, 6
(1999), 7-24.
214
Chapter 8
is in the percentage of exceptions, that is, the percentage of times the
actual loss exceeds VaR. The other is in the extent to which exceptions are
bunched. There are statistical tests to determine whether a VaR model
should be rejected because of the percentage of exceptions or the amount
of bunching. Regulators have rules for increasing the VaR multiplier
when market risk capital is calculated if they consider the results from
back testing over 250 days to be unsatisfactory.
Stress testing is an important complement to VaR calculations. It
considers scenarios that either have occurred in the past or are considered
possibilities for the future. Typically, the scenarios have a very low
probability of occurring under the models used for calculating VaR.
FURTHER READING
Artzner P., F. Delbaen, J.-M. Eber, and D. Heath, "Coherent Measures of
Risk," Mathematical Finance, 9 (1999): 203-228.
Basak, S., and A. Shapiro, "Value-at-Risk-Based Risk Management: Optimal
Policies and Asset Prices," Review of Financial Studies, 14, No. 2 (2001):
371-405.
Beder, T., "VaR: Seductive But Dangerous," Financial Analysts Journal, 51,
No. 5 (1995): 12-24.
Boudoukh, J., M. Richardson, and R. Whitelaw, "The Best of Both Worlds,"
Risk, May 1998: 64-67.
Dowd, K., Measuring Market Risk. 2nd edn. New York: Wiley, 2005.
Duffie, D., and J. Pan, "An Overview of Value at Risk," Journal of Derivatives,
4, No. 3 (Spring 1997): 7-49.
Hopper, G., "Value at Risk: A New Methodology for Measuring Portfolio
Risk," Business Review, Federal Reserve Bank of Philadelphia, July/August
1996: 19-29.
Hua P., and P. Wilmott, "Crash Courses," Risk, June 1997: 64-67.
Jackson, P., D. J. Maude, and W. Perraudin, "Bank Capital and Value at Risk,'
Journal of Derivatives, 4, No. 3 (Spring 1997): 73-90.
Jorion, P., Value at Risk. 2nd edn. New York: McGraw-Hill, 2001.
Longin, F.M., "Beyond the VaR," Journal of Derivatives, 8, No. 4 (Summer
2001): 36-48.
Marshall, C, and M. Siegel, "Value at Risk: Implementing a Risk Measurement
Standard," Journal of Derivatives, 4, No. 3 (Spring 1997): 91—111.
The VaR Measure
215
QUESTIONS AND PROBLEMS (Answers at End of Book)
8.1. What is the difference between expected shortfall and VaR? What is the
theoretical advantage of expected shortfall over VaR?
8.2. What is a spectral risk measure? What conditions must be satisfied by a
spectral risk measure for the Subadditivity condition in Section 8.3 to be
satisfied?
8.3. A fund manager announces that the fund's 1-month 95% VaR is 6% of
the size of the portfolio being managed. You have an investment of
$100,000 in the fund. How do you interpret the portfolio manager's
announcement?
8.4. A fund manager announces that the fund's one-month 95% expected
shortfall is 6% of the size of the portfolio being managed. You have an
investment of $100,000 in the fund. How do you interpret the portfolio
manager's announcement?
8.5. Suppose that each of two investments has a 0.9% chance of a loss of
$10 million, a 99.1% of a loss of $1 million, and zero probability of a gain.
The investments are independent of each other. (a) What is the VaR for
one of the investments when the confidence level is 99%? (b) What is the
expected shortfall for one of the investments when the confidence level is
99%? (c) What is the VaR for a portfolio consisting of the two investments
when the confidence level is 99%? (d) What is the expected shortfall for a
portfolio consisting of the two investments when the confidence level is
99%? (e) Show that in this example VaR does not satisfy the Subadditivity
condition whereas expected shortfall does.
8.6. Suppose that the change in the value of a portfolio over a 1-day time
period is normal with a mean of zero and a standard deviation of
$2 million, What is (a) the 1-day 97.5% VaR, (b) the 5-day 97.5% VaR,
and (c) the 5-day 99% VaR?
8.7. What difference does it make to your answers to (b) and (c) of Problem 8.6
if there is first-order daily autocorrelation with correlation parameter
equal to 0.16?
8.8. Explain carefully the differences between marginal VaR, incremental VaR,
and component VaR for a portfolio consisting of a number of assets.
8.9. Suppose that we back test a VaR model using 1,000 days of data. The VaR
confidence level is 99% and we observe 17 exceptions. Should we reject the
model at the 5% confidence level? Use a one-tailed test.
8.10. Explain what is meant by bunching.
8.11. Describe two ways extreme scenarios can be developed for stress testing.
8.12. Prove equation (8.3).
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Chapter 8
8.13. The back-testing rules of the Basel Committee can lead to questions about
a VaR model when there are 5 or more exceptions in 250 trials. What is the
chance of this if the VaR methodology is perfectly accurate?
ASSIGNMENT QUESTIONS
8.14. Suppose that each of two investments has a 4% chance of a loss of
$10 million, a 2% chance of a loss of $1 million, and a 94% chance of
a profit of $1 million. They are independent of each other. (a) What is the
VaR for one of the investments when the confidence level is 95%? (b) What
is the expected shortfall when the confidence level is 95%? (c) What is the
VaR for a portfolio consisting of the two investments when the confidence
level is 95%? (d) What is the expected shortfall for a portfolio consisting of
the two investments when the confidence level is 95%? (e) Show that, in
this example, VaR does not satisfy the Subadditivity condition whereas
expected shortfall does.
8.15. Suppose that daily changes for a portfolio have first-order correlation with
correlation parameter 0.12. The 10-day VaR, calculated by multiplying the
1-day VaR by
is $2 million. What is a better estimate of the VaR that
takes account of autocorrelation?
8.16. The probability that the loss from a portfolio will be greater than
$10 million in 1 month is estimated to be 5%. (a) What is the 1-month
99% VaR assuming the change in value of the portfolio is normally
distributed. (b) What is the 1-month 99% VaR assuming that the power
law described in Section 5.4 applies with = 3.
8.17. Suppose that we back test a VaR model using 1,000 days of data. The VaR
confidence level is 99% and we observe 15 exceptions. Should we reject the
model at the 5% confidence level. Use Kupiec's two-tailed test.
Market Risk VaR:
Historical Simulation
Approach
In this chapter and the next we cover the two main approaches for
calculating VaR for market risk. The approach we consider in this
chapter is known as historical simulation. This involves using historical
day-to-day changes in the values of market variables in a direct way to
estimate the probability distribution of the change in the value of the
current portfolio between today and tomorrow.
After describing the mechanics of the historical simulation approach,
we explain how to calculate the standard error of the VaR estimate and
how to modify the procedure to take into account the latest information
about volatility. We also describe how extreme value theory can be used
in conjuction with a historical simulation to improve VaR estimates and
to deal with situations where the VaR confidence level is very high.
9.1 THE METHODOLOGY
Historical simulation involves using past data as a guide to what will
happen in the future. Suppose that we want to calculate VaR for a
Portfolio using a one-day time horizon, a 99% confidence level, and
500 days of data. (The time horizon and confidence level are those
typically used for a market risk VaR calculation; 500 is a popular choice
for the number of days of data used.) The first step is to identify the
market variables affecting the portfolio. These will typically be exchange
218
Chapter 9
rates, equity prices, interest rates, and so on. We then collect data on the
movements in these market variables over the most recent 500 days. This
provides us with 500 alternative scenarios for what can happen between
today and tomorrow. Denote the first day for which we have data as
Day 0, the second day as Day 1, and so on. Scenario 1 is where the
percentage changes in the values of all variables are the same as they were
between Day 0 and Day 1; Scenario 2 is where they are the same as
between Day 1 and Day 2; and so on. For each scenario we calculate the
dollar change in the value of the portfolio between today and tomorrow.
This defines a probability distribution for daily changes in the value of
our portfolio. The first percentile of the distribution can be estimated as
the fifth worst outcome.1 The estimate of VaR is the loss when we are at
this first percentile point. We are 99% certain that we would not have
taken a loss greater than our VaR estimate if the changes in market
variables are a random sample from the last 500 days.
The historical simulation methodology is illustrated in Tables 9.1
and 9.2. Table 9.1 shows observations on market variables over the last
500 days. The observations are taken at some particular point in time
during the day (usually the close of trading). There are assumed to be a
total of 1,000 market variables.
Table 9.2 shows the values of the market variables tomorrow if their
percentage changes between today and tomorrow are the same as they
Table 9.1
Data for VaR historical simulation calculation.
Day
Market
variable 1
Market
variable 2
Market
variable 1,000
0
1
2
3
20.33
20.78
21.44
20.97
0.1132
0.1159
0.1162
0.1184
65.37
64.91
65.02
64.90
498
499
500
25.72
25.75
25.85
0.1312
0.1323
0.1343
62.22
61.99
62.10
1
There are alternatives here. A case could be made for using the fifth worst outcome or
the sixth worst outcome, or an average of the two, as the first percentile of the
distribution when there are 500 outcomes. In Excel's PERCENTILE function, when there
are n observations and k is an integer, the k/(n — 1) percentile is the observation ranked
k + 1. Other percentiles are calculated using linear interpolation.
VaR: Historical Simulation Approach
219
Table 9.2 Scenarios generated for tomorrow (Day 501) using data in Table 9.1.
Value of portfolio on Day 500 is $23.50 million
Scenario Market
number variable 1
Market
variable 2
1
2
3
26.42
26.67
25.28
0.1375
0.1346
0.1368
499
500
25.88
25.95
0.1354
0.1363
Market
..,. variable 1,000
••
••
Portfolio
Change
value
in value
($ millions) ($ millions)
61.66
62.21
61.99
23.71
23.12
22.94
0.21
-0.38
-0.56
61.87
62.21
23.63
22.87
0.13
-0.63
were between Day i — 1 and Day i for 1 i 500. The first row in
Table 9.2 shows the values of market variables tomorrow assuming their
percentage changes between today and tomorrow are the same as they
were between Day 0 and Day 1; the second row shows the values of
market variables tomorrow assuming their percentage changes are the
same as those between Day 1 and Day 2; and so on. The 500 rows in
Table 9.2 are the 500 scenarios considered.
Define vi as the value of a market variable on Day i and suppose that
today is Day n. The ith scenario assumes that the value of the market
variable tomorrow will be
In our example, n = 500. For the first variable, the value today,
is
25.85. In addition,
= 20.33 and = 20.78. It follows that the value of
the first market variable in the first scenario is
The penultimate column of Table 9.2 shows the value of the portfolio
tomorrow for each of the 500 scenarios. We suppose the value of the
Portfolio today is $23.50 million. This leads to the numbers in the final
column for the change in the value between today and tomorrow for all
the different scenarios. For Scenario 1 the change in value is +$210,000;
for Scenario 2 it is -$380,000; and so on.
We are interested in the one-percentile point of the distribution of
changes in the portfolio value. As indicated earlier, because there are a
220
Chapter 9
total of 500 scenarios in Table 9.2, we can estimate this as the fifth worst
number in the final column of the table. Alternatively, we can use extreme
value theory, which will be described later in the chapter. As mentioned in
Section 8.4, the ten-day VaR for a 99% confidence level is usually
calculated as
times the one-day VaR.
Each day the VaR estimate in our example would be updated using the
most recent 500 days of data. Consider, for example, what happens on
Day 501. We find out new values for all the market variables and are able
to calculate a new value for our portfolio.2 We then go through the
procedure we have outlined to calculate a new VaR. We use data on
the market variables from Day 1 to Day 501. (This gives us the required
500 observations on percentage changes in market variables; the Day 0
values of the market variables are no longer used.) Similarly, on Day 502,
we use data from Day 2 to Day 502 to determine VaR, and so on.
9.2 ACCURACY
The historical simulation approach estimates the distribution of portfolio
changes based on a finite number of observations of what happened in
the past. As a result, the estimates of quantiles of the distribution are not
perfectly accurate.
Kendall and Stuart describe how to calculate a confidence interval for
the quantile of a probability distribution when it is estimated from sample
data.3 Suppose that the q-quantile of the distribution is estimated as x.
The standard error of the estimate is
where n is the number of observations and f(x) is the probability density
function of the loss evaluated at x. The latter can be estimated by fitting
the empirical data to a standard distribution.
Example 9.1
Suppose we are interested in estimating the 0.01 quantile (= 1 percentile) of a
loss distribution from 500 observations so that n = 500 and q — 0.01. We can
estimate f{x) by approximating the actual empirical distribution with a
2
3
Note that the portfolio's composition may have changed between Day 500 and Day 501.
See M.G. Kendall and A. Stuart, The Advanced Theory of Statistics, Vol. 1:
Distribution Theory, 4th edn. London: Griffin, 1972.
VaR: Historical Simulation Approach
221
standard distribution. Suppose that the approximate empirical distribution is
normal with mean zero and standard deviation $10 million. Using Excel, the
0.01 quantile is NORMINV(0.01,0, 10), or 23.26. The value of f(x) is
NORMDIST(23.26, 0, 10, FALSE), or 0.0027. The standard error of the
estimate that is made is
If the estimate of the 0.01 quantile using historical simulation is $25 million,
a 95% confidence interval is from 25 — 1.96 x 1.67 to 25 + 1.96 x 1.67, that
is, from $21.7 million to $28.3 million.
As Example 9.1 illustrates, the standard error of a VaR estimated using
historical simulation tends to be quite high. It increases as the VaR
confidence level is increased. For example, if in Example 9.1 the VaR
confidence level had been 95% instead of 99%, the standard error would
be $0.95 million instead of $1.67 million. The standard error declines as
the sample size is increased—but only as the square root of the sample
size. If we quadrupled the sample size in Example 9.1 from 500 to 2,000
(i.e., from approximately two to approximately eight years of data), the
standard error halves from $1.67 million to about $0.83 million.
Additionally, we should bear in mind that historical simulation assumes
that the joint distribution of daily changes in market variables is stationary
through time. This is unlikely to be exactly true and creates additional
uncertainty about the value of VaR.
9.3 EXTENSIONS
In this section we cover a number of extensions of the basic historical
simulation methodology that we discussed in Section 9.1.
Weighting of Observations
The basic historical simulation approach assumes that each day in the past
is given equal weight. More formally, if we have observations for n day-today changes, each of them is given a weighting of \/n. Boudoukh et al.
suggest that more recent observations should be given more weight
because they are more reflective of current volatilities and current macroeconomic variables.4 The natural weighting scheme to use is one where
4
See J. Boudoukh, M. Richardson, and R. Whitelaw, "The Best of Both Worlds: A
Hybrid Approach to Calculating Value at Risk," Risk, 11 (May 1998), 64-67.
222
Chapter 9
weights decline exponentially. We used this in Section 5.6 when developing
the exponentially weighted moving average model for monitoring variance. Suppose that we are now at the end of day n. The weight assigned to
the change in the portfolio value between day n — i and day n — i + 1 is
times that assigned to the change between day n — i + 1 and day n — i + 2.
In order for the weights to add up to 1, the weight given to the change
between day n — i and n — i + 1 is
where n is the number of days. As
this weighting scheme
approaches the basic historical simulation approach, where all observations are given a weight of \/n (see Problem 9.2).
VaR is calculated by ranking the observations from the worst outcome
to the best. Starting at the worst outcome, weights are summed until the
required quantile of the distribution is reached. For example, if we are
calculating VaR with a 99% confidence level, we continue summing
weights until the sum just exceeds 0.01. We have then reached the 99%
VaR level. The best value of can be obtained by experimenting to see
which value back tests best. One disadvantage of the exponential weighting approach relative to the basic historical simulation approach is that the
effective sample size is reduced. However, we can compensate for this by
using a larger value of n. Indeed it is not really necessary to discard old
days as we move forward in time because they are given very little weight.
Incorporating Volatility Updating
Hull and White suggest a way of incorporating volatility updating into the
historical simulation approach. 5 A volatility updating scheme, such as
EWMA or GARCH(1,1) (both of which were described in Chapter 5) is
used in parallel with the historical simulation approach for all market
variables. Suppose that the daily volatility for a particular market variable estimated at the end of day i — 1 is
This is an estimate of the daily
volatility between the end of day i — 1 and the end of day i. Suppose it is
now day n. The current estimate of the volatility of the market variable is
This applies to the time period between today and tomorrow, which
is the time period over which we are calculating VaR.
Suppose that
is twice
This means that we estimate the daily
5
See J. Hull and A. White, "Incorporating Volatility Updating into the Historical
Simulation Method for Value-at-Risk," Journal of Risk, 1, No. 1 (Fall 1998), 5-19.
VaR: Historical Simulation Approach
223
volatility of this particular market variable to be twice as great today as on
day i — 1. This means that we expect to see changes between today and
tomorrow that are twice as big as changes between day i — 1 and day i.
When carrying out the historical simulation and creating a sample of what
could happen between today and tomorrow based on what happened
between day i — 1 and day i, it therefore makes sense to multiply the latter
by 2. In general, when this approach is used, the expression in equation (9.1) for the value of a market variable under the ith scenario becomes
Each market variable is handled in the same way. This approach takes
account of volatility changes in a natural and intuitive way and produces
VaR estimates that incorporate more current information. The VaR
estimates can be greater than any of the historical losses that would have
occurred for our current portfolio on the days we consider. Hull and White
produce evidence using exchange rates and stock indices to show that this
approach is superior to traditional historical simulation and to the exponential weighting scheme described earlier. More complicated models can
be developed where observations are adjusted for the latest information on
correlations as well as for the latest information on volatilities.
Bootstrap Method
The bootstrap method is another variation on the basic historical simulation approach. It involves creating a set of changes in the portfolio value
based on historical movements in market variables in the usual way. We
then sample with replacement from these changes to create many new
similar data sets. We calculate the VaR for each of the new data sets. Our
95% confidence interval for VaR is the range between the 2.5 and the
97.5 percentile point of the distribution of the VaRs calculated from the
data sets.
Suppose, for example, that we have 500 days of data. We could sample
with replacement 500,000 times from the data to obtain 1,000 different
sets of 500 days of data. We calculate the VaR for each set. We then rank
the VaRs. Suppose that the 25th largest VaR is $5.3 million and the 475th
largest VaR is $8.9 million. The 95% confidence limit for VaR is
$5-3 million to $8.9 million. Usually the 95% confidence range calculated
for VaR using the bootstrap method is less than that calculated using the
Procedure in Section 9.2.
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Chapter 9
9.4 EXTREME VALUE THEORY
In Section 5.4 we introduced the power law and explained that it can be
used to estimate the tails of a wide range of distributions. We now provide
the theoretical underpinnings for the power law and present more sophisticated estimation procedures than those used in Section 5.4. The term
used to describe the science of estimating the tails of a distribution is
extreme value theory. In this section we show how extreme value theory
can be used to improve VaR estimates and to deal with situations where
the VaR confidence level is very high. It provides a way of smoothing and
extrapolating the tails of an empirical distribution.
The Key Result
A key result in extreme value theory was proved by Gnedenko in 1943.6
This concerns the properties of the tails of a wide range of different
probability distributions.
Suppose that F(x) is the cumulative distribution function for a variable
x and that u is a value of x in the right-hand tail of the distribution. The
probability that x lies between u and u + y (y > 0) is F(u + y) — F(u).
The probability that x is greater than u is 1 — F(u). Define Fu(y) as the
probability that x lies between u and u + y conditional on x > u. This is
The variable Fu(y) defines the right tail of the probability distribution. It
is the cumulative probability distribution for the amount by which x
exceeds u given that it does exceed u.
Gnedenko's result states that, for a wide class of distributions F(x), the
distribution of Fu(y) converges to a generalized Pareto distribution as the
threshold u is increased. The generalized Pareto distribution is
The distribution has two parameters that have to be estimated from the
data. These are and
The parameter is the shape parameter and
6
See D.V. Gnedenko, "Sur la distribution limite du terme d'une serie aleatoire," AnnMath., 44 (1943), 423-453.
VaR: Historical Simulation Approach
225
determines the heaviness of the tail of the distribution. The parameter is
a scale parameter.
When the underlying variable x has a normal distribution, = 0.7 As
the tails of the distribution become heavier, the value of increases. For
most financial data, is positive and in the range 0.1 to 0.4.8
Estimating
and
The parameters
and
can be estimated using maximum-likelihood
methods (see Section 5.9 for a discussion of these methods). The
probability density function
of the cumulative distribution in
equation (9.3) is calculated by differentiating with respect to y. It is
We first choose a value for u. This could be a value close to the 95 percentile
point of the empirical distribution. We then rank the observations on x
from the highest to the lowest and focus our attention on those observations for which x > u. Suppose there are nu such observations and they are
xi (1 i nu). The likelihood function (assuming that
0) is
Maximizing this function is the same as maximizing its logarithm:
Standard numerical procedures can be used to find the values of
that maximize this expression.
and
7
When
= 0, the generalized Pareto distribution becomes
One of the properties of the distribution in equation (9.3) is that the kth moment E(xk)
x is infinite for
For a normal distribution, all moments are finite. When
= 0.25, only the first three moments are finite; when = 0.5, only the first moment is
finite; and so on.
226
Chapter 9
Estimating the Tail of the Distribution
The probability that x > u + y conditional that x > u is
The
probability that x > u is 1 — F(u). The unconditional probability that
x > u + y is therefore
If n is the total number of observations, an estimate of 1 — F{u) calculated from the empirical data is nu/n. The unconditional probability that
x > u + y is therefore
This means that our estimator of the tail of the cumulative probability
distribution of x when x is large is
Equivalence to the Power Law
If we set
equation (9.5) reduces to
so that the probability of the variable being greater than x is
where
and
This shows that equation (9.5) is consistent with the power
law introduced in Section 5.4.
The Left Tail
The analysis so far has assumed that we are interested in the right tail of
the probability distribution. If we are interested in the left tail, we can use
the methodology just presented on the variable —x.
227
VaR: Historical Simulation Approach
Calculation of VaR
To calculate VaR with a confidence level of q it is necessary to solve the
equation
From equation (9.5), we have
so that
9.5 APPLICATION
We now illustrate the results in the previous section using data for daily
returns on the S&P 500 between July 11, 1988, and July 10, 1998. During
this period the total number of observations, n, on the daily return were
2,256 and the observations ranged from —6.87% to +5.12%. We consider
the left tail of the distribution of returns. This means that in the equations
given above the variable x is the negative of the daily return on the
S&P 500. We choose a value for u equal to 0.02. There were a total of
28 returns less than —2%. This means than nu = 28. The returns and the
x-values are shown in the first two columns of Table 9.3. The third
column shows the value of
for particular values of and
(The test values of and in Table 9.3
are = 0.2 and = 0.01.) The sum of the numbers in the third column is
the log-likelihood function in equation (9.4). Once we have set up the
spreadsheet, we search for the best-fit values of and that maximize the
tog-likelihood function.9 It turns out that these are
= 0.3232,
= 0.0055
and maximum log-likelihood is 108.48.
9
The Solver routine in Excel works well provided that the spreadsheet is set up so that
the values being searched for are similar in magnitude. In this example, we could set up
the spreadsheet to search for and
228
Chapter 9
Table 9.3
Estimation of extreme value theory parameters.
Daily return
xi
-0.068667
-0.061172
-0.036586
-0.034445
-0.031596
-0.030827
-0.029979
-0.029654
-0.029084
-0.027283
-0.025859
-0.025364
-0.024675
-0.024000
-0.023485
-0.023397
-0.023234
-0.022675
-0.022542
-0.022343
-0.022249
-0.022020
-0.021813
-0.021025
-0.020843
-0.020625
-0.020546
-0.020243
0.068667
0.061172
0.036586
0.034445
0.031596
0.030827
0.029979
0.029654
0.029084
0.027283
0.025859
0.025364
0.024675
0.024000
0.023485
0.023397
0.023234
0.022675
0.022542
0.022343
0.022249
0.022020
0.021813
0.021025
0.020843
0.020625
0.020546
0.020243
Trial estimatesof EVT parameters
0.2
0.01
0.5268
1.0008
2.8864
3.0825
3.3537
3.4291
3.5133
3.5460
3.6035
3.7893
3.9403
3.9937
4.0689
4.1434
4.2009
4.2108
4.2291
4.2925
4.3076
4.3304
4.3412
4.3676
4.3915
4.4835
4.5049
4.5306
4.5400
4.5761
106.1842
VaR: Historical Simulation Approach
229
Suppose that we wish to estimate the probability that x will be less than
0.04. From equation (9.5), this is
This means that we estimate the probability that the daily return will be
less than - 4 % to be 1 - 0.9989 = 0.0011. (This is more accurate than an
estimate obtained by counting observations.) The probability that x will
be less than 0.06 is similarly 0.9997. This means that we estimate the
probability that the daily return will be less than - 6 % to be
1 - 0.9997 = 0.0003.
From equation (9.6) the value of the one-day 99% VaR for a portfolio
where $1 million is invested in the S&P 500 is $1 million times
or $21,200. More generally, our estimate of the one-day 99% VaR for a
portfolio invested in the S&P 500 is 2.12% of the portfolio value.
Choice of u
A natural question is how the results depend on the choice of u. In our
example the values of and do depend on u, but the estimates of F(x)
remain roughly the same. For example, if we choose u = 0.015, the bestfit values of and are 0.264 and 0.0046, respectively. The estimate for
F(x) when x = 0.04 and x = 0.06 are 0.9989 and 0.9997 (much the same
as before). The estimate of VaR also does not change too much provided
that the confidence level is not too high. The one-day 99% VaR for an
investment in the S&P 500 when u = 0.015 is 2.13% (compared with
2.12% when u = 0.02) of the value of the portfolio.
SUMMARY
Historical simulation is a very popular approach for estimating VaR. It
involves creating a database consisting of the daily movements in all
market variables over a period of time. The first simulation trial assumes
that the percentage change in each market variable is the same as that
on the first day covered by the database; the second simulation trial
230
Chapter 9
assumes that the percentage changes are the same as those on the second
day; and so on. The change in the portfolio value,
is calculated for
each simulation trial, and VaR is calculated as the appropriate percentile
of the probability distribution of
The procedure assumes that the
future will in some sense be like the past. The standard error for a VaR
that is estimated using historical simulation tends to be quite high. The
higher the VaR confidence level required, the higher the standard error.
There are a number of extensions of the basic historical simulation
approach. The weights given to observations can be allowed to decrease
exponentially as we look further and further into the past. Volatility
updating schemes can be used to take account of differences between the
volatilities of market variables today and their volatilities at different times
during the period covered by the historical data.
Extreme value theory is a way of smoothing the tails of the probability
distribution of portfolio daily changes calculated using historical simulation. It leads to estimates of VaR that reflect the whole shape of the tail of
the distribution, not just the positions of a few losses in the tails. Extreme
value theory can also be used to estimate VaR when the VaR confidence
level is very high. For example, even if we have only 500 days of data, it
could be used to come up an estimate of VaR for a VaR confidence level
of 99.9%.
FURTHER READING
Boudoukh, J., M. Richardson, and R. Whitelaw, "The Best of Both Worlds,"
Risk, 11 (May 1998): 64-67.
Embrechts, P., C. Kluppelberg, and T. Mikosch, Modeling Extremal Events for
Insurance and Finance. New York: Springer, 1997.
Hendricks, D., "Evaluation of Value-at-Risk Models Using Historical Data,'
Economic Policy Review, Federal Reserve Bank of New York, Vol. 2 (April
1996): 39-69.
Hull, J. C, and A. White, "Incorporating Volatility Updating into the Historical
Simulation Method for Value at Risk," Journal of Risk, 1, No. 1 (Fall 1998):
5-19.
McNeil, A. J., "Extreme Value Theory for Risk Managers," in Internal Modeling
and CAD II, Risk Books, 1999. See also: www.math.ethz.ch/~mcneil.
Neftci, S.N., "Value at Risk Calculations, Extreme Events and Tail
Estimation," Journal of Derivatives, 1, No. 3 (Spring 2000): 23-38.
VaR:
Historical
Simulation
Approach
231
QUESTIONS A N D PROBLEMS (Answers at End of Book)
9.1. What assumptions are being made when VaR is calculated using the
historical simulation approach and 500 days of data.
9.2. Show that when
approaches 1, the weighting scheme in Section 9.3
approaches the basic historical simulation approach.
9.3. Calculate the 1-day 99% VaR on February 10, 2006, for a £100 million
portfolio invested in the FTSE 100 Index. Use the previous 1,000 days of
data (available on the author's website) and historical simulation.
9.4. Repeat Problem 9.3 using the exponential weighting scheme in Section 9.3
with = 0.99.
9.5. Repeat Problem 9.3 using the volatility updating scheme discussed in
Section 9.3. Use EWMA with = 0.94 to update volatilities. Assume that
the volatility is initially equal to the standard deviation of daily returns
calculated from the whole sample.
9.6. How does extreme value theory modify your answer to Problem 9.3. Try
values of u equal to 0.005, 0.01, and 0.015.
9.7. Suppose we estimate the 1-day 95% VaR from 1,000 observations as
$5 million. By fitting a standard distribution to the observations, the
probability density function of the loss distribution at the 95% point is
estimated to be 0.01 when losses are measured in millions. What is the
standard error of the VaR estimate?
ASSIGNMENT QUESTIONS
9.8. Values for the NASDAQ composite index during the 1,500 days preceding
March 10, 2006, can be downloaded from the author's website. Calculate
the 1-day 99% VaR on March 10, 2006, for a $10 million portfolio invested
in the index using (a) the basic historical simulation approach, (b) the
exponential weighting scheme in Section 9.3 with = 0.995, (c) the volatility
updating scheme in Section 9.3 with = 0.94 (assume that the volatility is
initially equal to the standard deviation of daily returns calculated from the
whole sample), (d) extreme value theory with u = 0.03, (e) a model where
daily returns are assumed to be normally distributed (use both an approach
where observations are given equal weights and the EWMA approach with
= 0.94 to estimate the standard deviation of daily returns). Discuss the
reasons for the differences between the results you get.
9.9. Suppose that a 1-day 97.5% VaR is estimated as $13 million from 2,000
observations. The observation on the 1-day changes are approximately
232
Chapter 9
normal with mean 0 and standard deviation $6 million. Estimate a 99%
confidence interval for the VaR estimate.
Market Risk VaR:
Model-Building
Approach
The main alternative to the historical simulation approach for estimating
VaR for market risk is what is known as the model-building approach or
the Variance-covariance approach. In this approach, we assume a model
for the joint distribution of changes in market variables and use historical data to estimate the model parameters.
The model-building approach is based on ideas pioneered by Harry
Markowitz. We used these ideas for assessing the risk-return trade-offs
in portfolios of stocks in Section 1.1. They can also be used to calculate
VaR. Estimates of the current levels of the variances and covariances of
market variables are made using the approaches described in Chapters 5
and 6. If the probability distributions of the daily percentage changes in
market variables are assumed to be normal and the dollar change in the
value of the portfolio is assumed to be linearly dependent on percentage
changes in market variables, VaR can be obtained very quickly.
In this chapter we explain the model-building approach and show how
it can be used for portfolios consisting of stocks, bonds, forward contracts, and interest rate swaps. We discuss attempts to extend it to
situations where the portfolio is not linearly dependent on the market
Variables and to situations where the distributions of daily percentage
changes in market variables are not normal. Finally, we evaluate the
strengths and weaknesses of both the model-building approach and the
historical simulation approach.
234
Chapter 10
10.1 THE BASIC METHODOLOGY
In option pricing, volatility is normally measured as "volatility per year".
When using the model-building approach to calculate VaR, we usually
measure the volatility of an asset as "volatility per day". As explained in
Section 5.1, the relationship between the volatility per day and the volatility
per year is
where
is the volatility per year and
is the corresponding volatility
per day. This equation shows that the daily volatility is about 6% of
annual volatility.
As also pointed out in Chapter 5,
is approximately equal to the
standard deviation of the percentage change in the asset price in one day.
For the purposes of calculating VaR, we assume exact equality. We define
the daily volatility of an asset price (or any other variable) as equal to the
standard deviation of the percentage change in one day.
Single-Asset Case
We now consider how VaR is calculated using the model-building
approach in a very simple situation where the portfolio consists of a
position in a single stock. The portfolio we consider is one consisting of
$10 million in shares of Microsoft. We suppose that the time horizon is
ten days and the VaR confidence level is 99%, so that we are interested in
the loss level over ten days that we are 99% confident will not be
exceeded. Initially, we consider a one-day time horizon.
We assume that the volatility of Microsoft is 2% per day (corresponding to about 32% per year). Because the size of the position is $10 million,
the standard deviation of daily changes in the value of the position is 2%
of $10 million, or $200,000.
It is customary in the model-building approach to assume that the
expected change in a market variable over the time period considered is
zero. This is not exactly true, but it is a reasonable assumption. The
expected change in the price of a market variable over a short time period
is generally small when compared to the standard deviation of the
change. Suppose, for example, that Microsoft has an expected return of
20% per annum. Over a one-day period, the expected return is 0.20/252,
or about 0.08%, whereas the standard deviation of the return is 2%. Over
a ten-day period, the expected return is 0.08 x 10, or about 0.8%'
whereas the standard deviation of the return is
or about 6.3%.
VaR Model-Building Approach
235
So far, we have established that the change in the value of the
portfolio of Microsoft shares over a one-day period has a standard
deviation of $200,000 and (at least approximately) a mean of zero. We
assume that the change is normally distributed.1 Since N(—2.33) = 0.01,
this means that there is a 1% probability that a normally distributed
variable will decrease in value by more than 2.33 standard deviations.
Equivalently, it means that we are 99% certain that a normally distributed variable will not decrease in value by more than 2.33 standard
deviations. The one-day 99% VaR for our portfolio consisting of a
$10 million position in Microsoft is therefore
2.33 x 200,000 = $466,000
Assuming that the changes in Microsoft's stock price on successive days
are independent, the N-day VaR is calculated as
times the one-day
VaR. The ten-day 99% VaR for Microsoft is therefore
Consider next a portfolio consisting of a $5 million position in AT&T,
and suppose the daily volatility of AT&T is 1% (approximately 16% per
year). A similar calculation to that for Microsoft shows that the standard
deviation of the change in the value of the portfolio in one day is
5,000,000 x 0.01 = 50,000
Assuming that the change is normally distributed, the one-day 99% VaR is
50,000x2.33 = $116,500
and the ten-day 99% VaR is
Two-Asset Case
Now consider a portfolio consisting of both $10 million of Microsoft
shares and $5 million of AT&T shares. We suppose that the returns on
the two shares have a bivariate normal distribution with a correlation
of 0.3. A standard result in statistics tells us that, if two variables X and Y
have standard deviations equal to
and
with the coefficient of
1
We could assume that the price of Microsoft is lognormal tomorrow. Since one day is
such a short period of time, this is almost indistinguishable from the assumption we do
make—that the change in the stock price between today and tomorrow is normal.
Chapter 10
236
correlation between them being equal to
X + Y is given by
the standard deviation of
To apply this result, we set X equal to the change in the value of the
position in Microsoft over a one-day period and Y equal to the change in
the value of the position in AT&T over a one-day period, so that
The standard deviation of the change in the value of the portfolio
consisting of both stocks over a one-day period is therefore
The mean change is assumed to be zero. The change is normally distributed. So the one-day 99% VaR is therefore
220,227 x 2.33 = $513,129
The ten-day 99% VaR is
times this or $1,622,657.
The Benefits of Diversification
In the example we have just considered:
1. The ten-day 99% VaR for the portfolio of Microsoft shares is
$1,473,621.
2. The ten-day 99% VaR for the portfolio of AT&T shares is $368,405.
3. The ten-day 99% VaR for the portfolio of both Microsoft and
AT&T shares is $1,622,657.
The amount
(1,473,621 + 368,405) - 1,622,657 = $219,369
represents the benefits of diversification. If Microsoft and AT&T were
perfectly correlated, the VaR for the portfolio of both Microsoft and
AT&T would equal the VaR for the Microsoft portfolio plus the VaR for
the AT&T portfolio. Less than perfect correlation leads to some of the
risk being "diversified away".2
2
VaR reflects the benefits of diversification when the distribution of the portfolio-value
changes is normal. As we saw in Section 8.3, VaR does not always reflect the benefits of
diversification. The VaR of two portfolios can be greater than the sum of their separate
VaRs.
VaR: Model-Building Approach
237
10.2 THE LINEAR MODEL
The examples we have just considered are simple illustrations of the use
of the linear model for calculating VaR. Suppose that we have a portfolio worth P consisting of n assets with an amount
being invested in
asset i (1 i n). Define
as the return on asset i in one day. The
dollar change in the value of our investment in asset i in one day is
and
where
is the dollar change in the value of the whole portfolio in
one day.
In the example considered in the previous section, $10 million was
invested in the first asset (Microsoft) and $5 million was invested in the
second asset (AT&T) so that (in millions of dollars)
=10,
=5,
and
If we assume that the
in equation (10.1) are multivariate normal,
is normally distributed. To calculate VaR, we therefore need to
calculate only the mean and standard deviation of
We assume, as
discussed in the previous section, that the expected value of each
is
zero. This implies that the mean of
is zero.
To calculate the standard deviation of
we define as the daily
volatility of the ith asset and
as the coefficient of correlation between
returns on asset i and asset j . 3 This means that
is the standard
deviation of
and
is the coefficient of correlation between
and
The variance of
which we will denote by is given by
This equation can also be written as
3
The
are sometimes calculated using a factor model (see Section 6.3).
238
Chapter 10
The standard deviation of the change over N days is
and the 99%
VaR for an N-day time horizon is
In the example considered in the previous section,
= 0.02, =0.01,
and
= 0.3. As already noted,
= 10 and
= 5, so that
= 102 x 0.022 + 52 x 0.012 + 2 x 10 x 5 x 0.3 x 0.02 x 0.01 = 0.0485
and
= 0.220. This is the standard deviation of the change in the
portfolio value per day (in millions of dollars). The ten-day 99% VaR
is 2.33 x 0.220 x
= $1,623 million. This agrees with the calculation
in the previous section.
10.3 HANDLING INTEREST RATES
It is not possible to define a separate market variable for every single
bond price or interest rate to which a company is exposed. Some
simplifications are necessary when the model-building approach is used.
One possibility is to assume that only parallel shifts in the yield curve
occur. It is then necessary to define only one market variable: the size of
the parallel shift. The changes in the value of bond portfolio can then be
calculated using the approximate duration relationship in equation (4.15):
where P is the value of the portfolio,
is the change in P in one day, D is
the modified duration of the portfolio, and
is the parallel shift in one
day. This approach gives a linear relationship between
and
but it
does not usually give enough accuracy because the relationship is not
exact and does not take account of nonparallel shifts in the yield curve.
The procedure usually followed is to choose as market variables the
prices of zero-coupon bonds with standard maturities: 1 month, 3 months,
6 months, 1 year, 2 years, 5 years, 7 years, 10 years, and 30 years. For the
purposes of calculating VaR, the cash flows from instruments in the
portfolio are mapped into cash flows occurring on the standard maturity
dates.
Consider a $1 million position in a Treasury bond lasting 0.8 years
that pays a coupon of 10% semiannually. A coupon is paid in 0.3 years
and 0.8 years and the principal is paid in 0.8 years. This bond is
therefore in the first instance regarded as a $50,000 position in 0.3-year
zero-coupon bond plus a $1,050,000 position in a 0.8-year zero-coupon
VaRs Model-Building Approach
239
bond. The position in the 0.3-year bond is then replaced by an equivalent position in 3-month and 6-month zero-coupon bonds and the
position in the 0.8-year bond is replaced by an equivalent position in
6-month and 1-year zero-coupon bonds. The result is that the position
in the 0.8-year coupon-bearing bond is for VaR purposes regarded as a
position in zero-coupon bonds having maturities of 3 months, 6 months,
and 1 year. This procedure is known as cash-flow mapping.
Illustration of Cash-Flow Mapping
We now illustrate how cash-flow mapping works by continuing with the
example we have just introduced. It should be emphasized that the procedure we use is just one of several that have been proposed.
Consider first the $1,050,000 that will be received in 0.8 years. We
suppose that zero rates, daily bond price volatilities, and correlations
between bond returns are as shown in Table 10.1. The first stage is to
interpolate between the 6-month rate of 6.0% and the 1-year rate of 7.0%
to obtain a 0.8-year rate of 6.6% (annual compounding is assumed for all
rates). The present value of the $1,050,000 cash flow to be received in
0.8 years is
We also interpolate between the 0.1 % volatility for the 6-month bond
and the 0.2% volatility for the 1-year bond to get a 0.16% volatility for
the 0.8-year bond.
Table 10.1
Data to illustrate cash-flow mapping procedure.
Maturity
Zero rate
(% with ann. comp.):
Bond price volatility
(% per day):
Correlation between
daily returns
3-month bond
6-month bond
1-year bond
3-month
bond
6-month
bond
1-year
bond
5.50
6.00
7.00
0.06
0.10
0.20
3-month
bond
6-month
bond
1-year
bond
1.0
0.9
0.6
0.9
1.0
0.7
0.6
0.7
1.0
240
Chapter 10
Suppose we allocate of the present value to the 6-month bond and
of the present value to the 1-year bond. Using equation (10.2) and
matching variances, we obtain
This is a quadratic equation that can be solved in the usual way to give
= 0.320337. This means that 32.0337% of the value should be allocated
to a 6-month zero-coupon bond and 67.9663% of the value should be
allocated to a 1-year zero-coupon bond. The 0.8-year bond worth $997,662
is therefore replaced by a 6-month bond worth
997,662 x 0.320337 = $319,589
and a 1-year bond worth
997,662 x 0.679663 = $678,074
This cash-flow-mapping scheme has the advantage that it preserves both
the value and the variance of the cash flow. Also, it can be shown that the
weights assigned to the two adjacent zero-coupon bonds are always
positive.
For the $50,000 cash flow received at time 0.3 years, we can carry out
similar calculations (see Problem 10.7). It turns out that the present value
of the cash flow is $49,189. This can be mapped to a position worth
$37,397 in a 3-month bond and a position worth $11,793 in a 6-month
bond.
The results of the calculations are summarized in Table 10.2. The
0.8-year coupon-bearing bond is mapped to a position worth $37,397
in a 3-month bond, a position worth $331,382 in a 6-month bond, and a
position worth $678,074 in a 1-year bond. Using the volatilities and
correlations in Table 10.1, equation (10.2) gives the variance of the
Table 10.2
The cash-flow-mapping result.
$50,000
received
in 0.3 years
Position in 3-month bond ($):
Position in 6-month bond ($):
Position in 1-year bond ($):
37,397
11,793
$1,050,000
received
in 0.8 years
319,589
678,074
Total
37,397
331,382
678,074
VaR: Model-Building Approach
241
change in the price of the 0.8-year bond with n = 3,
= 37,397,
= 331,382,
=678,074;
=0.0006,
=0.001 and
=0.002;
and
=0.9,
=0.6,
= 0.7. This variance is 2,628,518. The standard deviation of the change in the price of the bond is therefore
= 1,621.3. Because we are assuming that the bond is the only
instrument in the portfolio, the ten-day 99% VaR is
or about $11,950.
Principal Components Analysis
As we explained in Section 4.10, a principal components analysis (PCA)
can be used to reduce the number of deltas that are calculated for
movements in a zero-coupon yield curve. A PCA can also be used (in
conjunction with cash-flow mapping) to handle interest rates when VaR
is calculated using the model-building approach. For any given portfolio, we can convert a set of delta exposures, such as those given in Table 4.11, into a delta exposure to the first PCA factor, a delta
exposure to the second PCA factor, and so on. This is done in
Section 4.10. In the example in Table 4.11 the exposure to the first
factor is calculated as —0.08 and the exposure to the second factor is
calculated as —4.40. (The first two factors capture over 90% of the
variation in interest rates.) Suppose that f1 and f2 are the factor scores.
The change in the portfolio value is approximately
The factor scores in a PCA are uncorrelated. From Table 4.10 their
standard deviations of the first two factors are 17.49 and 6.05. The
standard deviation of
is therefore
The one-day 99% VaR is therefore 26.66 x 2.33 = 62.12. Note that the
Portfolio we are considering has very little exposure to the first factor and
significant exposure to the second factor. Using only one factor would
significantly understate VaR (see Problem 10.9). The duration-based
method for handling interest rates would also significantly understate
VaR as it considers only parallel shifts in the yield curve.
242
Chapter 10
10.4 APPLICATIONS OF THE LINEAR MODEL
The simplest application of the linear model is to a portfolio with no
derivatives consisting of positions in stocks, bonds, foreign exchange, and
commodities. In this case the change in the value of the portfolio is
linearly dependent on the percentage changes in the prices of the assets
comprising the portfolio. Note that, for the purposes of VaR calculations,
all asset prices are measured in the domestic currency. The market
variables considered by a large bank in the United States are therefore
likely to include the value of the Nikkei 225 Index measured in dollars, the
price of a ten-year sterling zero-coupon bond measured in dollars, and
so on.
Examples of derivatives that can be handled by the linear model are
forward contracts on foreign exchange and interest rate swaps. Suppose a
forward foreign exchange contract matures at time T. It can be regarded as
the exchange of a foreign zero-coupon bond maturing at time T for a
domestic zero-coupon bond maturing at time T. Therefore, for the purposes of calculating VaR, the forward contract is treated as a long position
in the foreign bond combined with a short position in the domestic bond.
(As just mentioned, the foreign bond is valued in the domestic currency.)
Each bond can be handled using a cash-flow-mapping procedure so that it
is a linear combination of bonds with standard maturities.
Consider next an interest rate swap. This can be regarded as the
exchange of a floating-rate bond for a fixed-rate bond. The fixed-rate
bond is a regular coupon-bearing bond (see Appendix B). The floatingrate bond is worth par just after the next payment date. It can be regarded
as a zero-coupon bond with a maturity date equal to the next payment
date. The interest rate swap therefore reduces to a portfolio of long and
short positions in bonds and can be handled using a cash-flow-mapping
procedure.
10.5 THE LINEAR MODEL AND OPTIONS
We now consider how the linear model can be used when there are
options. Consider first a portfolio consisting of options on a single stock
whose current price is S. Suppose that the delta of the position (calculated
in the way described in Chapter 3) is 4 Because is the rate of change of
4
In Chapter 3 we denote the delta and gamma of a portfolio by and In this section
and the next, we use the lower case Greek letters and to avoid overworking
VaR: Model-Building Approach
243
the value of the portfolio with S, it is approximately true that
or
(10.3)
(10.3)
where
is the dollar change in the stock price in one day and
usual, the dollar change in the portfolio in one day. We define
return on the stock in one day, so that
It follows that an approximate relationship between
and
is, as
as the
is
When we have a position in several underlying market variables that
includes options, we can derive an approximate linear relationship between
and the
similarly. This relationship is
(10.4)
where is the value of the ith market variable and is the delta of the
portfolio with respect to the ith. market variable. This is equation (10.1):
(10.5)
with
Equation (10.5) can therefore be used to calculate the
standard deviation of
Example 10.1
A portfolio consists of options on Microsoft and AT&T. The options on
Microsoft have a delta of 1,000, and the options on AT&T have a delta of
20,000. The Microsoft share price is $120, and the AT&T share price is $30.
From equation (10.4), it is approximately true that
or
where
and
are the returns from Microsoft and AT&T in one day
244
Chapter 10
and
is the resultant change in the value of the portfolio. (The portfolio is
assumed to be equivalent to an investment of $120,000 in Microsoft and
$600,000 in AT&T.) Assuming that the daily volatility of Microsoft is 2%
and the daily volatility of AT&T is 1 % and the correlation between the daily
changes is 0.3, the standard deviation of
(in thousands of dollars) is
Because N(-1.65) = 0.05, the five-day 95% VaR is
1.65 x
x 7,099 = $26,193
Weakness of the Model
When a portfolio includes options, the linear model is an approximation.
It does not take account of the gamma of the portfolio. As discussed in
Chapter 3, delta is defined as the rate of change of the portfolio value
with respect to an underlying market variable and gamma is defined as
the rate of change of the delta with respect to the market variable.
Gamma measures the curvature of the relationship between the portfolio
value and an underlying market variable.
Figure 10.1 shows the impact of a nonzero gamma on the probability
distribution of the value of the portfolio. When gamma is positive, the
probability distribution tends to be positively skewed; when gamma is
negative, it tends to be negatively skewed. Figures 10.2 and 10.3 illustrate
the reason for this result. Figure 10.2 shows the relationship between the
value of a long call option and the price of the underlying asset. A long
call is an example of an option position with positive gamma. The figure
shows that, when the probability distribution for the price of the underlying asset at the end of one day is normal, the probability distribution
Figure 10.1 Probability distribution for value of portfolio: (a) positive gamma;
(b) negative gamma.
VaR: Model-Building Approach
245
Figure 10.2 Translation of normal probability distribution for an asset
into probability distribution for value of a long call on that asset.
for the option price is positively skewed. Figure 10.3 shows the relationship between the value of a short call position and the price of the
underlying asset. A short call position has a negative gamma. In this
case, we see that a normal distribution for the price of the underlying
asset at the end of one day gets mapped into a negatively skewed
distribution for the value of the option position.5
The VaR for a portfolio is critically dependent on the left tail of the
probability distribution of the portfolio value. For example, when the
confidence level used is 99%, the VaR is the value in the left tail below
which only 1% of the distribution resides. As indicated in Figures 10.1a
and 10.2, a positive gamma portfolio tends to have a less heavy left tail
than the normal distribution. If we assume the distribution is normal,
we will tend to calculate a VaR that is too high. Similarly, as indicated
in Figures 10.1b and 10.3, a negative gamma portfolio tends to have a
heavier left tail than the normal distribution. If we assume the distributaon is normal, we will tend to calculate a VaR that is too low.
5
The normal distribution is a good approximation to the lognormal distribution for
short time periods.
246
Chapter 10
Figure 10.3 Translation of normal probability distribution for an asset
into probability distribution for value of a short call on that asset.
10.6 THE QUADRATIC MODEL
For a more accurate estimate of VaR than that given by the linear model,
we can use both delta and gamma measures to relate
to the
Consider a portfolio dependent on a single asset whose price is S.
Suppose and are the delta and gamma of the portfolio. As indicated
in Chapter 3, a Taylor Series expansion gives
as an improvement over the approximation in equation (10.3).6 Setting
reduces this to
6
A fuller Taylor series expansion suggests the approximation
when terms of higher order than At are ignored. In practice, the
that it is usually ignored.
term is so small
VaR: Model-Building Approach
247
In this case, we have
where is the daily volatility of the variable.
For a portfolio with n underlying market variables, with each instrument in the portfolio being dependent on only one of the market
variables, equation (10.6) becomes
where is the value of the ith market variable, and and are the delta
and gamma of the portfolio with respect to the ith market variable. When
some of the individual instruments in the portfolio are dependent on
more than one market variable, this equation takes the more general form
where
is a "cross gamma", defined as
Equation (10.7) is not as easy to work with as equation (10.5), but it can
be used to calculate moments for
Cornish-Fisher Expansion
A result in statistics known as the Cornish-Fisher expansion can be used
to estimate quantiles of a probability distribution from its moments. We
illustrate this by showing how the first three moments can be used to
Produce a VaR estimate that takes account of the Skewness of the
Probability distribution. Define
and
as the mean and standard
deviation of
so that
248
The skewness
Chapter 10
of the probability distribution of
is defined as
Using the first three moments of
the Cornish-Fisher expansion
estimates the q-quantile of the distribution of
as
where
and zq is q-quantile of the standard normal distribution.
Example 10.2
Suppose that for a certain portfolio we calculate
= —0.2,
= 2.2, and
= —0.4, and we are interested in the 0.01 quantile (q = 0.01). In this case,
zq = —2.33. If we assume that the probability distribution of
is normal,
then the 0.01 quantile is
- 0 . 2 - 2 . 3 3 x 2 . 2 = -5.326
In other words, we are 99% certain that
When we use the Cornish-Fisher expansion to adjust for skewness and set
q = 0.01, we obtain
so that the 0.01 quantile of the distribution is
-0.2 - 2.625 x 2.2 = -5.976
Taking account of Skewness, therefore, changes the VaR from 5.326 to 5.976.
10.7 MONTE CARLO SIMULATION
As an alternative to the approaches described so far, we can implement
the model-building approach using Monte Carlo simulation to generate
the probability distribution for
Suppose we wish to calculate a oneday VaR for a portfolio. The procedure is as follows:
1. Value the portfolio today in the usual way using the current values
of market variables.
VaR Model-Building Approach
249
2. Sample once from the multivariate normal probability distribution
7
of the
3. Use the sampled values of the
to determine the value of each
market variable at the end of one day.
4. Revalue the portfolio at the end of the day in the usual way.
5. Subtract the value calculated in Step 1 from the value in Step 4 to
determine a sample
6. Repeat Steps 2 to 5 many times to build up a probability
distribution for
The VaR is calculated as the appropriate percentile of the probability
distribution of
Suppose, for example, that we calculate 5,000 different sample values of
in the way just described. The one-day 99% VaR
is the 50th worst outcome; the one-day 95% VaR is the 250th worst
outcome; and so on. 8 The N-day VaR is usually assumed to be the oneday VaR multiplied by
The drawback of Monte Carlo simulation is that it tends to be
computationally slow because a company's complete portfolio (which
might consist of hundreds of thousands of different instruments) has to
be revalued many times.10 One way of speeding things up is to assume
that equation (10.7) describes the relationship between
and the
We can then jump straight from Step 2 to Step 5 in the Monte Carlo
simulation and avoid the need for a complete revaluation of the portfolio.
This is sometimes referred to as the partial simulation approach.
10.8 USING DISTRIBUTIONS THAT ARE NOT NORMAL
When Monte Carlo simulation is used, there are ways of extending the
model-building approach so that market variables are no longer assumed
to be normal. One possibility is to assume that the variables have a
multivariate t-distribution. As indicated in Figures 6.4 and 6.5, this has
7
One way of doing so is given in Chapter 6.
As in the case of historical simulation, extreme value theory can be used to "smooth the
tails", so that better estimates of extreme percentiles are obtained.
9
This is only approximately true when the portfolio includes options, but it is the
assumption that is made in practice for most VaR calculation methods.
10
An approach for limiting the number of portfolio revaluations is proposed in
F. Jamshidian and Y. Zhu, "Scenario Simulation Model: Theory and Methodology,"
Finance and Stochastics, 1 (1997), 43-67.
8
250
Chapter 10
the effect of giving a higher value to the probability that extreme values
for several variables occur simultaneously.
We can assume any set of distributions for the
in conjunction with
a copula model.11 Suppose, for example, that we use a Gaussian copula
model. As explained in Chapter 6, this means that, when the changes
in market variables are transformed on a percentile-to-percentile basis to
normally distributed variables ui the ui are multivariate normal. We can
follow the five steps given earlier except that Step 2 is changed and a step
is inserted between Steps 2 and 3 as follows:
2. Sample once from the multivariate probability distribution for
the ui.
2a. Transform each ui to
on a percentile-to-percentile basis.
If a bank has already implemented the Monte Carlo simulation approach
for calculating VaR assuming percentage changes in market variables are
normal, it should be relatively easy to modify calculations to implement
the approach we describe here. Just before the portfolio is valued it is
necessary to insert a line or two of code into the computer program to do
the transformation in Step 2a. The marginal distributions of the
can
be calculated by fitting a more general distribution than the normal
distribution to empirical data.
10.9 MODEL BUILDING vs. HISTORICAL SIMULATION
In the last chapter and this one, we have discussed two methods for
estimating VaR: the historical simulation approach and the model-building approach. The advantages of the model-building approach are that
results can be produced very quickly and it can easily be used in
conjunction with volatility and correlation updating schemes such as
those described in Chapters 5 and 6. As mentioned in Section 9.3,
volatility updating can be incorporated into the historical simulation
approach—but in a rather more artificial way. The main disadvantage
of the model-building approach is that (at least in the simplest version of
the approach) it assumes that the market variables have a multivariate
normal distribution. In practice, daily changes in market variables often
have distributions that are quite different from normal (see, for example,
Table 5.2). A user of the model-building approach is hoping that some
See J. Hull and A. White, "Value at Risk When Daily Changes Are Not Normally
Distributed," Journal of Derivatives, 5, No. 3 (Spring 1998), 9-19.
VaR: Model-Building Approach
251
form of the central limit theorem of statistics applies to the portfolio, so
that the probability distribution of daily gains/losses on the portfolio is
normally distributed—even though the gains/losses on the component
parts of the portfolio are not normally distributed.
The historical simulation approach has the advantage that historical
data determines the joint probability distribution of the market variables.
It is also easier to handle interest rates in a historical simulation because
on each trial a complete zero-coupon yield curve for both today and
tomorrow can be calculated. The somewhat messy cash-flow-mapping
procedure described in Section 10.3 is avoided. The main disadvantage
of historical simulation is that it is computationally much slower than the
model-building approach. It is sometimes necessary to use an approximation such as equation (10.7) when using the historical simulation approach. This is because a full revaluation for each of the 500 different
scenarios is not possible.12
The model-building approach is often used for investment portfolios.
(It is after all closely related to the popular Markowitz mean-variance
method of portfolio analysis.) It is less commonly used for calculating the
VaR for the trading operations of a financial institution. This is because,
as explained in Chapter 3, financial institutions like to maintain their
deltas with respect to market variables close to zero. Neither the linear
model nor the quadratic model work well when deltas are low and
portfolios are nonlinear.
SUMMARY
Whereas historical simulation lets the data determine the joint probability
distribution of daily percentage changes in market variables, the modelbuilding approach assumes a particular form for this distribution. The
most common assumption is that percentage changes in the variables have
a multivariate normal distribution. For situations where the change in the
value of the portfolio is linearly dependent on percentage changes in the
market variables, VaR can be calculated exactly in a straightforward way.
In other situations approximations are necessary. One approach is to use a
quadratic approximation for the change in the value of the portfolio as a
12
This is particularly likely to be the case if Monte Carlo simulation is the numerical
Procedure used by the financial institution to value a deal. Monte Carlo simulations
within historical simulations lead to extremely time-consuming computations.
252
Chapter 10
function of percentage changes in the market variables. Another (much
slower) approach is to use Monte Carlo simulation.
The model-building approach is frequently used for investment portfolios. It is less popular for the trading portfolios of financial institutions
because it does not work well when deltas are low.
FURTHER READING
Frye, J., "Principals of Risk: Finding VAR through Factor-Based Interest Rate
Scenarios," in VAR: Understanding and Applying Value at Risk, pp. 275-288.
London: Risk Publications, 1997.
Hull, J.C., and A. White, "Value at Risk When Daily Changes in Market
Variables Are Not Normally Distributed," Journal of Derivatives, 5 (Spring
1998): 9-19.
Jamshidian, F., and Y. Zhu, "Scenario Simulation Model: Theory and
Methodology," Finance and Stochastics, 1 (1997): 43-67.
Rich, D., "Second Generation VaR and Risk-Adjusted Return on Capital,"
Journal of Derivatives, 10, No. 4 (Summer 2003): 51-61.
QUESTIONS AND PROBLEMS (Answers at End of Book)
10.1. Consider a position consisting of a $100,000 investment in asset A and a
$100,000 investment in asset B. Assume that the daily volatilities of both
assets are 1 % and that the coefficient of correlation between their returns
is 0.3. What is the 5-day 99% VaR for the portfolio?
10.2. Describe three ways of handling interest-rate-dependent instruments
when the model building approach is used to calculate VaR.
10.3. Explain how an interest rate swap is mapped into a portfolio of zerocoupon bonds with standard maturities for the purposes of a VaR
calculation.
10.4. A financial institution owns a portfolio of options on the USD/GBP
exchange rate. The delta of the portfolio is 56.0. The current exchange
rate is 1.5000. Derive an approximate linear relationship between the
change in the portfolio value and the percentage change in the exchange
rate. If the daily volatility of the exchange rate is 0.7%, estimate the tenday 99% VaR.
10.5. Suppose you know that the gamma of the portfolio in Problem 10.4 is 16.2.
How does this change your estimate of the relationship between the change
in the portfolio value and the percentage change in the exchange rate?
VaR:
Model-Building
Approach
253
10.6. Suppose that the 5-year rate is 6%, the seven year rate is 7% (both
expressed with annual compounding), the daily volatility of a 5-year zerocoupon bond is 0.5%, and the daily volatility of a 7-year zero-coupon
bond is 0.58%. The correlation between daily returns on the two bonds is
0.6. Map a cash flow of $1,000 received at time 6.5 years into a position
in a 5-year bond and a position in a 7-year bond. What cash flows in 5
and 7 years are equivalent to the 6.5-year cash flow?
10.7. Verify that the 0.3-year zero-coupon bond in the cash-flow-mapping
example in Table 10.2 is mapped into a $37,397 position in a 3-month
bond and a $11,793 position in a 6-month bond.
10.8. Suppose that the daily change in the value of a portfolio is, to a good
approximation, linearly dependent on two factors, calculated from a
principal components analysis. The delta of a portfolio with respect to
the first factor is 6 and the delta with respect to the second factor is -4.
The standard deviations of the factor are 20 and 8, respectively. What is
the 5-day 90% VaR?
10.9. The text calculates a VaR estimate for the example in Table 4.11
assuming two factors. How does the estimate change if you assume
(a) one factor and (b) three factors.
10.10. A bank has a portfolio of options on an asset. The delta of the options is
-30 and the gamma is - 5 . Explain how these numbers can be interpreted.
The asset price is 20 and its volatility is 1 % per day. Using the quadratic
model calculate the first three moments of the change in the portfolio
value. Calculate a 1-day 99% VaR using (a) the first two moments and
(b) the first three moments.
10.11. Suppose that in Problem 10.10 the vega of the portfolio is —2 per 1%
change in the annual volatility. Derive a model relating the change in the
portfolio value in 1 day to delta, gamma, and vega. Explain, without doing
detailed calculations, how you would use the model to estimate a VaR.
10.12. Explain why the linear model can provide only approximate estimates of
VaR for a portfolio containing options.
10.13. Some time ago a company entered into a forward contract to buy
£1 million for $1.5 million. The contract now has 6 months to maturity.
The daily volatility of a 6-month zero-coupon sterling bond (when
its price is translated to dollars) is 0.06% and the daily volatility of
a 6-month zero-coupon dollar bond is 0.05%. The correlation between
returns from the two bonds is 0.8. The current exchange rate is 1.53.
Calculate the standard deviation of the change in the dollar value of the
forward contract in 1 day. What is the 10-day 99% VaR? Assume that
the 6-month interest rate in both sterling and dollars is 5% per annum
with continuous compounding.
254
Chapter 1()
ASSIGNMENT QUESTIONS
10.14. Consider a position consisting of a $300,000 investment in gold and a
$500,000 investment in silver. Suppose that the daily volatilities of these
two assets are 1.8% and 1.2% respectively, and that the coefficient of
correlation between their returns is 0.6. What is the 10-day 97.5% VaR
for the portfolio? By how much does diversification reduce the VaR?
10.15. Consider a portfolio of options on a single asset. Suppose that the delta
of the portfolio is 12, the value of the asset is $10, and the daily volatility
of the asset is 2%. Estimate the 1-day 95% VaR for the portfolio from
the delta.
10.16. Suppose you know that the gamma of the portfolio in Problem 10.15 is
-2.6. Derive a quadratic relationship between the change in the portfolio
value and the percentage change in the underlying asset price in 1 day.
(a) Calculate the first three moments of the change in the portfolio value.
(b) Using the first two moments and assuming that the change in the
portfolio is normally distributed, calculate the 1-day 95% VaR for the
portfolio, (c) Use the third moment and the Cornish-Fisher expansion to
revise your answer to (b).
10.17. A company has a long position in a 2-year bond and a 3-year bond as
well as a short position in a 5-year bond. Each bond has a principal of
$100 and pays a 5% coupon annually. Calculate the company's exposure
to the 1-year, 2-year, 3-year, 4-year, and 5-year rates. Use the data in
Tables 4.8 and 4.9 to calculate a 20-day 95% VaR on the assumption that
rate changes are explained by (a) one factor, (b) two factors, and (c) three
factors. Assume that the zero-coupon yield curve is flat at 5%.
10.18. A company has a position in bonds worth $6 million. The modified
duration of the portfolio is 5.2 years. Assume that only parallel shifts
in the yield curve can take place and that the standard deviation of the
daily yield change (when yield is measured in percent) is 0.09. Use the
duration model to estimate the 20-day 90% VaR for the portfolio.
Explain carefully the weaknesses of this approach to calculating VaR
Explain two alternatives that give more accuracy.
10.19. A bank has written a call option on one stock and a put option on
another stock. For the first option the stock price is 50, the strike price is
51, the volatility is 28% per annum, and the time to maturity is 9 months.
For the second option the stock price is 20, the strike price is 19, the
volatility is 25% per annum, and the time to maturity is 1 year. Neither
stock pays a dividend, the risk-free rate is 6% per annum, and the
correlation between stock price returns is 0.4. Calculate a 10-day 99%
VaR (a) using only deltas, (b) using the partial simulation approach, and
(c) using the full simulation approach.
Credit Risk:
Estimating Default
Probabilities
This is the first of three chapters concerned with credit risk. Credit risk
arises from the possibility that borrowers, bond issuers, and counterparties in derivatives transactions may default. As explained in Chapter 7,
regulators have for a long time required banks to keep capital for credit
risk. In Basel II, banks can, with approval from bank supervisors, use
their own estimates of default probabilities to determine the amount of
capital they are required to keep. This has led banks to devote a lot of
resources to developing better ways of estimating default probabilities.
In this chapter we discuss a number of different approaches to estimating
default probabilities and explain the key difference between risk-neutral
and real-world estimates. The material we cover will be used in both
Chapters 12 and 13. In Chapter 12 we examine the nature of the credit
risk in over-the-counter derivatives transactions and discuss the calculation
of credit value at risk. In Chapter 13 we cover credit derivatives.
11.1 CREDIT RATINGS
Rating agencies such as Moody's and S&P are in the business of providing ratings describing the creditworthiness of corporate bonds. Using the
Moody's system, the best rating is Aaa. Bonds with this rating are
considered to have almost no chance of defaulting. The next best rating
is Aa. Following that comes A, Baa, Ba, B, and Caa. Only bonds with
256
Chapter \\
ratings of Baa or above are considered to be investment grade. The S&P
ratings corresponding to Moody's Aaa, Aa, A, Baa, Ba, B, and Caa are
AAA, AA, A, BBB, BB, B, and CCC, respectively. To create finer rating
measures, Moody's divides the Aa rating category into Aal, Aa2, and
Aa3; it divides A into Al, A2 and A3; and so on. Similarly S&P divides
its AA rating category into AA+, AA, and AA-; it divides its A rating
category into A+, A, and A - ; and so on. (Only the Aaa category for
Moody's and the AAA category for S&P are not subdivided.)
A credit rating is designed to provide information about default
probabilities. As such one might expect frequent changes in a company's
credit rating as positive and negative information reaches the market.1 In
fact, ratings change relatively infrequently. When rating agencies assign
ratings, one of their objectives is ratings stability. For example, they want
to avoid ratings reversals where a company is downgraded and then
upgraded a few weeks later. Ratings therefore change only when there
is reason to believe that a long-term change in the company's creditworthiness has taken place. The reason for this is that bond traders are
major users of ratings. Often they are subject to rules governing what the
credit ratings of the bonds they hold must be. If these ratings changed
frequently they might have to do a large amount of trading (and incur
high transactions costs) just to satisfy the rules.
A related point is that rating agencies try to "rate through the cycle".
Suppose that the economy exhibits a downturn and this has the effect of
increasing the probability of a company defaulting in the next six months,
but makes very little difference to the company's cumulative probability
of defaulting over the next three to five years. A rating agency would not
change the company's credit rating.
Internal Credit Ratings
Most banks have procedures for rating the creditworthiness of their
corporate and retail clients. This is a necessity. The ratings published
by rating agencies are only available for relatively large corporate clientsMany small and medium-sized businesses do not issue publicly traded
bonds and therefore are not rated by rating agencies. As explained in
Chapter 7, the internal ratings based (IRB) approach in Basel II allows
banks to use their internal ratings in determining the probability of
1
In theory, a credit rating is an attribute of a bond issue, not a company. However, in
most cases all bonds issued by a company have the same rating. A rating is therefor
often referred to as an attribute of a company.
Credit Risk: Estimating Default Probabilities
257
default, PD. Under the advanced IRB approach, they are also allowed to
use their own internal procedures for estimating the loss given default,
LGD, the exposure at default, EAD, and the maturity, M.
Internal ratings based approaches for PD typically involve profitability
ratios, such as return on assets, and balance-sheet ratios, such as the
current ratio and the debt-to-equity ratio. Banks recognize that it is cash
rather than profits that is necessary to repay a loan. They typically take
the financial information provided by a company and convert it to the
type of cash-flow statement that allows them to estimate how easy it will
be for a company to service its debt.
Altman's Z-Score
Edward Altman has pioneered the use of accounting ratios to predict
default. In 1968 he developed what has become known as the Z-score.2
Using a statistical technique known as discriminant analysis, he attempted
to predict defaults from five accounting ratios:
X1:
X2:
X3:
X4:
X5:
Working capital/Total Assets
Retained earnings/Total assets
Earnings before interest and taxes/Total assets
Market value of equity/Book value of total liabilities
Sales/Total assets
The Z-score was calculated as
If the Z-score is greater than 3.0, the company is unlikely to default. If it is
between 2.7 and 3.0, we should be "on alert". If it is between 1.8 and 2.7,
there is a good chance of default. If it is less than 1.8, the probability of a
financial embarrassment is very high. The Z-score was calculated from a
sample of 66 publicly traded manufacturing companies. Of these, 33 failed
within one year and 33 did not fail. The model proved very accurate when
tested on a sample of firms different from that used to obtain equation
(11.l). Both Type I errors (companies that were predicted not to go
bankrupt but did do so) and Type II errors (companies that were predicted
to go bankrupt, but did not do so) were small.3 Variations on the model
2
See E.I. Altman, "Financial Ratios, Discriminant Analysis, and the Prediction of
Corporate Bankruptcy," Journal of Finance, 23, No. 4 (September 1968), 589-609.
Clearly Type I errors are much more costly to the lending department of a commercial
bank than Type II errors.
258
Chapter 11
have been developed for manufacturing companies that are not publicly
traded and nonmanufacturing companies.
Example 11.1
Consider a company for which working capital is 170,000, total assets are
670,000, earnings before interest and taxes is 60,000, sales are 2,200,000, the
market value of equity is 380,000, total liabilities is 240,000, and retained
earnings is 300,000. In this case X1 = 0.254, X2 = 0.448, X3 = 0.0896,
X4 = 1.583, and X5 = 3.284. The Z-score is
1.2 x 0.254 + 1.4 x 0.448 + 3.3 x 0.0896 + 0.6 x 1.583 + 0.999 x 3.284 = 5.46
The Z-score indicates that the company is not in danger of defaulting in the
near future.
11.2 HISTORICAL DEFAULT PROBABILITIES
Table 11.1 is typical of the data that is produced by rating agencies. It
shows the default experience through time of companies that started with
a certain credit rating. For example, Table 11.1 shows that a bond issue
with an initial credit rating of Baa has a 0.20% chance of defaulting by
the end of the first year, a 0.57% chance of defaulting by the end of the
second year, and so on. The probability of a bond defaulting during a
particular year can be calculated from the table. For example, the
probability that a bond initially rated Baa will default during the second
year of its life is 0.57 - 0.20 = 0.37%.
Table 11.1 shows that, for investment-grade bonds, the probability of
default in a year tends to be an increasing function of time. (For example,
Table 11.1
Average cumulative default rates (%), 1970-2003
(Source: Moody's).
Rating
Term (years)
1
Aaa
Aa
A
Baa
Ba
B
Caa
2
3
4
5
7
10
15
0.00 0.00 0.00 0.04 0.12 0.29 0.62
1.21
0.02 0.03 0.06 0.15 0.24 0.43 0.68
1.51
0.02 0.09 0.23 0.38 0.54 0.91
1.59
2.94
0.20 0.57
1.03
1.62 2.16 3.24 5.10 9.12
1.26 3.48 6.00 8.59 11.17 15.44 21.01 30.88
6.21 13.76 20.65 26.66 31.99 40.79 50.02 59.21
23.65 37.20 48.02 55.56 60.83 69.36 77.91 80.23
20
1.55
2.70
5.24
12.59
38.56
60.73
80.23
Credit Risk: Estimating Default Probabilities
259
the probability of an A-rated bond defaulting during years 1, 2, 3, 4, and 5
are 0.02%, 0.07%, 0.14%, 0.15%, and 0.16%, respectively.) This is
because the bond issuer is initially considered to be creditworthy, and
the more time that elapses, the greater the possibility that its financial
health will decline. For bonds with a poor credit rating, the probability of
default is often a decreasing function of time. (For example, the probabilities that a Caa-rated bond will default during years 1, 2, 3, 4, and 5 are
23.65%, 13.55%, 10.82%, 7.54%, and 5.27%, respectively.) The reason
here is that, for a bond with a poor credit rating, the next year or two may
be critical. If the issuer survives this period, its financial health is likely to
have improved.
Default Intensities
From Table 11.1 we can calculate the probability of a Caa bond defaulting during the third year as 48.02 - 37.20 = 10.82%. We will refer to this
as the unconditional default probability. It is the probability of default
during the third year as seen at time zero. The probability that the Caarated bond will survive until the end of year 2 is 100 - 37.20 = 62.80%.
The probability that it will default during the third year, conditional on
no earlier default, is therefore 0.1082/0.6280, or 17.23%.
The 17.23% we have just calculated is for a one-year time period. By
considering the probability of default between times t and
conditional on no earlier default, we obtain what is known as the default
intensity or hazard rate at time t. The default intensity
at time t is
defined so that
is the probability of default between time t and
conditional on no default between time 0 and time t. If V(t) is the
cumulative probability of the company surviving to time t (i.e., no default
by time t), then
Taking limits, we obtain
from which we get
or
where
is the average default intensity between time zero and time t.
260
Chapter 11
Define Q(t) as the probability of default by time t. It follows that
or
11.3 RECOVERY RATES
When a company goes bankrupt, those that are owed money by the
company file claims against the assets of the company.4 Sometimes there
is a reorganization in which these creditors agree to a partial payment of
their claims. In other cases the assets are sold by the liquidator and the
proceeds are used to meet the claims as far as possible. Some claims
typically have priorities over other claims and are met more fully.
The recovery rate for a bond is normally defined as the bond's market
value immediately after a default as a percent of its face value. It equals one
minus the loss given default, LGD. Table 11.2 provides historical data on
average recovery rates for different categories of bonds in the United
States. It shows that senior secured debtholders had an average recovery
rate of 57.4 cents per dollar of face value, while junior subordinated
debtholders had an average recovery rate of only 28.9 cents per dollar of
face value.
Recovery rates are significantly negatively correlated with default rates.
Moody's looked at average recovery rates and average default rates each
Table 11.2 Recovery rates on corporate bonds as a
percent of face value, 1982-2004. Source: Moody's.
Class
Senior secured
Senior unsecured
Senior subordinated
Subordinated
Junior subordinated
4
Average recovery
rate (%)
57.4
44.9
39.1
32.0
28.9
In the United States, the claim made by a bondholder is the bond's face value plus
accrued interest.
Credit Risk: Estimating Default Probabilities
261
year between 1983 and 2004. It found that the following relationship
provides a good fit to the data: 5
Average recovery rate = 0.52 — 6.9 x Average default rate
This relationship means that a bad year for the default rate is usually
doubly bad because it is accompanied by a low recovery rate. For
example, when the average default rate in a year is only 0.1%, we expect
the recovery rate to be relatively high at 51.3%. When it is relatively high
at 3%, we expect the recovery rate to be 31.3%.
11.4 ESTIMATING DEFAULT PROBABILITIES FROM
BOND PRICES
The probability of default for a company can be estimated from the prices
of bonds it has issued. The usual assumption is that the only reason a
corporate bond sells for less than a similar risk-free bond is the possibility
of default.6
Consider first an approximate calculation. Suppose that a bond yields
200 basis points more than a similar risk-free bond and that the expected
recovery rate in the event of a default is 40%. The holder of a corporate
bond must be expecting to lose 200 basis points (or 2% per year) from
defaults. Given the recovery rate of 40%, this leads to an estimate of the
probability of a default per year conditional on no earlier default of
0.02/(1 - 0.4), or 3.33%. In general,
where h is the default intensity per year, s is the spread of the
corporate bond yield over the risk-free rate, and R is the expected
recovery rate.
5
See D. T. Hamilton, P. Varma, S. Ou, and R. Cantor, "Default and Recovery Rates of
Corporate Bond Issuers, 1920-2004," Moody's Investor's Services, January 2005. The R2
of the regression is 0.65. The correlation is also identified and discussed in E. I. Altman,
B. Brady, A. Resti, and A. Sironi, "The Link between Default and Recovery Rates:
Implications for Credit Risk Models and Procyclicality," Working Paper, New York
University, 2003.
6
We discuss this point later. The assumption is not perfect. In practice, the price of a
corporate bond is also affected by its liquidity. The lower the liquidity, the lower the
price.
262
Chapter 11
A More Exact Calculation
For a more exact calculation, suppose that the corporate bond we have
been considering lasts for five years, provides a coupon of 6% per annum
(paid semiannually), and yields 7% per annum (with continuous compounding). The yield on a similar default-free bond is 5% (with continuous
compounding). The yields imply that the price of the corporate bond is
95.34 and the price of the default-free bond is 104.09. The expected loss
from default over the five-year life of the bond is therefore 104.09 - 95.34,
or $8.75. Suppose that the probability of default per year (assumed in this
simple example to be the same each year) is Q. Table 11.3 calculates the
expected loss from default in terms of Q on the assumption that defaults
can happen at times 0.5, 1.5, 2.5, 3.5, and 4.5 years (immediately before
coupon payment dates). Risk-free rates for all maturities are assumed to
be 5% (with continuous compounding).
To illustrate the calculations, consider the 3.5-year row in Table 11.3.
The expected value of the default-free bond at time 3.5 years (calculated
using forward interest rates) is
Given the definition of recovery rates in the previous section, the amount
recovered if there is a default is 40, so that the loss is 104.34 - 40, or
$64.34. The present value of this loss is 54.01 and the expected loss is
therefore 54.01 Q.
The total expected loss is 288.48(2. Setting this equal to 8.75, we obtain a
value for Q equal to 3.03%. The calculations we have given assume that
Table 11.3
Calculation of loss from default on a bond in terms of the default
probabilities per year, Q. Notional principal = $100.
Time
(years)
Default
probability
Recovery
amount
($)
Default-free
value ($)
Loss
($)
Discount
factor
0.5
1.5
2.5
3.5
4.5
Q
Q
Q
Q
Q
40
40
40
40
40
106.73
105.97
105.17
104.34
103.46
66.73
65.97
65.17
64.34
63.46
0.9753
0.9277
0,8825
0.8395
0.7985
Total
PV of
expectea
loss ($)
65.08Q
61.20Q
57.52Q
54.01Q
50.67Q
288.48Q
Credit Risk: Estimating Default Probabilities
263
the default probability is the same each year and that defaults take place
once a year. We can extend the calculations to assume that defaults can
take place more frequently. Furthermore, instead of assuming a constant
unconditional probability of default, we can assume a constant default
intensity or a particular pattern for the variation of default probabilities
with time.
With several bonds we can estimate several parameters describing the
term structure of default probabilities. Suppose, for example, we have
bonds maturing in 3, 5, 7, and 10 years. We could use the first bond to
estimate a default probability per year for the first three years, the second
to estimate a default probability per year for years 4 and 5, the third to
estimate a default probability for years 6 and 7, and the fourth to estimate
a default probability for years 8, 9, and 10 (see Problems 11.11 and
11.17). The approach is analogous to the bootstrap procedure we discussed in Chapter 4 for calculating a zero-coupon yield curve.
The Risk-Free Rate
A key issue when bond prices are used to estimate default probabilities is
the meaning of the terms "risk-free rate" and "risk-free bond". In equation (11.3) the spread s is the excess of the corporate bond yield over the
yield on a similar risk-free bond. In Table 11.3 the default-free value of the
bond must be calculated using risk-free rates. The benchmark risk-free rate
that is usually used in quoting corporate bond yields is the yield on similar
Treasury bonds (e.g., a bond trader might quote the yield on a particular
corporate bond as being a spread of 250 basis points over Treasuries).
As discussed in Section 4.4, traders usually use LIBOR/swap rates as
proxies for risk-free rates when valuing derivatives. Traders also use
LIBOR/swap rates as risk-free rates when calculating default probabilities. For example, when they determine default probabilities from bond
Prices, the spread s in equation (11.3) is the spread of the bond yield over
the LIBOR/swap rate. Also, the risk-free discount rates used in the
calculations such as those in Table 11.3 are LIBOR/swap zero rates.
Credit default swaps (which will be discussed in Chapter 13) can be
used to imply the risk-free rate assumed by traders. The rate used appears
to be approximately equal to the LIBOR/swap rate minus 10 basis points
on average.7 This estimate is plausible. As explained in Section 4.4, the
7
See J. Hull, M. Predescu, and A. White, "The Relationship between Credit Default
swap Spreads, Bond Yields, and Credit Rating Announcements," Journal of Banking and
Finance, 28 (November 2004), 2789-2811.
264
Chapter 11
credit risk in a swap rate is the credit risk from making a series of sixmonth loans to AA-rated counterparties and 10 basis points is a reasonable default risk premium for an AA-rated six-month instrument.
Asset Swaps
In practice, traders often use asset swap spreads as a way of extracting
default probabilities from bond prices. This is because asset swap spreads
provide a direct estimate of the spread of bond yields over the LIBOR/
swap curve.
To explain how asset swaps work, consider the situation where an asset
swap spread for a particular bond is quoted as 150 basis points. There are
three possible situations:
1. The bond sells for its par value of 100. The swap then involves one
side (Company A) paying the coupon on the bond and the other
side (Company B) paying LIBOR plus 150 basis points. 8
2. The bond sells below its par value, say, for 95. The swap is then
structured so that, in addition to the coupons, Company A pays $5
per $100 of notional principal at the outset.
3. The underlying bond sells above par, say, for 108. The swap is then
structured so that Company B makes a payment of $8 per $100 of
principal at the outset. After that, Company A pays the bond's
coupons and Company B pays LIBOR plus 150 basis points.
The effect of all this is that the present value of the asset swap spread is
the amount by which the price of the corporate bond is exceeded by the
price of a similar risk-free bond, where the risk-free rate is assumed to be
given by the LIBOR/swap curve (see Problem 11.12). Consider again the
example in Table 11.3 where the LIBOR/swap zero curve is flat at 5%.
Suppose that instead of knowing the bond's price we know that the asset
swap spread is 150 basis points. This means that the amount by which the
value of the risk-free bond exceeds the value of the corporate bond is the
present value of 150 basis points per year for five years. Assuming
semiannual payments, this is $6.55 per $100 of principal.
The total loss in Table 11.3 would be set equal to $6.55 in this case. This
means that the default probability per year, Q, would be 6.55/288.48,
or 2.27%.
8
Note that it is the promised coupons that are exchanged. The exchanges take place
regardless of whether the bond defaults.
Credit Risk: Estimating Default Probabilities
265
11.5 COMPARISON OF DEFAULT PROBABILITY
ESTIMATES
The default probabilities estimated from historical data are much less than
those derived from bond prices. Table 11.4 illustrates this.9 It shows, for
companies that start with a particular rating, the seven-year average annual
default intensity calculated from (a) historical data and (b) bond prices.
The calculation of default intensities using historical data is based on
equation (11.2) and Table 11.1. From equation (11.2), we have
where
is the average default intensity (or hazard rate) by time t
and Q(t) is the cumulative probability of default by time t. The values
of Q{1) are taken directly from Table 11.1. Consider, for example, an
A-rated company. The value of Q{7) is 0.0091. The average seven-year
default intensity is therefore
or 0.13%.
The calculations using bond prices are based on equation (11.3) and
bond yields published by Merrill Lynch. The results shown are averages
between December 1996 and July 2004. The recovery rate is assumed to be
40% and, for the reasons discussed in the previous section, the risk-free
interest rate is assumed to be the seven-year swap rate minus 10 basis
points. For example, for A-rated bonds the average Merrill Lynch yield was
Table 11.4
Rating
Aaa
Aa
A
Baa
Ba
B
Caa
9
Seven-year average default intensities (% per annum).
Historical default
intensity
Default intensity
from bonds
Ratio
Differeice
0.04
0.06
0.13
0.47
2.40
7.49
16.90
0.67
0.78
1.28
2.38
5.07
9.02
21.30
16.8
13.0
9.8
5.1
2.1
1.2
1.3
0.63
0.72
1.15
1.91
2.67
1.53
4.40
Tables 11.4 and 11.5 are taken from J. Hull, M. Predescu, and A. White, "Bond Prices,
Default Probabilities, and Risk Premiums," Journal of Credit Risk, 1, No. 2 (2004), 53-60.
266
Chapter 11
Table 11.5
Rating
Aaa
Aa
A
Baa
Ba
B
Caa
Expected excess return on bonds (basis points).
Bond yield
spread over
Treasuries
Spread of riskfree rate over
Treasuries
Spread for
historical
defaults
Expected
excess
return
83
90
120
186
347
585
1321
43
43
43
43
43
43
43
2
4
8
28
144
449
1014
38
43
69
115
160
93
264
6.274%. The average swap rate was 5.605%, so that the average risk free
rate was 5.505%. This gives the average seven-year default probability as
or 1.28%.
Table 11.4 shows that the ratio of the default probability backed out of
bond prices to the default probability calculated from historical data is
high for investment-grade companies and tends to decline as the credit
quality declines. By contrast, the difference between the two default
probabilities tends to increase as credit quality declines.
Table 11.5 gives another way of looking at these results. It shows the
excess return over the risk-free rate (still assumed to be the seven-year swap
rate minus 10 basis points) earned by investors in bonds with different
credit ratings. Consider again an A-rated bond. The average spread over
Treasuries is 120 basis points. Of this, 43 basis points represent the average
spread between seven-year Treasuries and our proxy for the risk-free rate.
A spread of 8 basis points is necessary to cover expected defaults. (This
equals the real-world probability of default from Table 11.4 times 1 minus
the assumed recovery rate of 0.4.) This leaves an expected excess return
(after expected defaults have been taken into account) of 69 basis points.
Tables 11.4 and 11.5 show that a large percentage difference between
default probability estimates translates into a relatively small expected
excess return on the bond. For Aaa-rated bonds the ratio of the two
default probabilities is 16.8, but the expected excess return is only 38 basis
points. The expected return tends to increase as credit quality declines.
10
The results for B-rated bonds in Tables 11.4 and 11.5 run counter to the overall pattern.
Credit Risk: Estimating Default Probabilities
Business Snapshot 11.1
267
Risk-Neutral Valuation
The single most important idea in the valuation of derivatives is risk-neutral
valuation. It shows that we can value a derivative by
1. Assuming that all investors are risk neutral
2. Calculating expected cash flows
3. Discounting the cash flows at the risk-free rate
As a simple example of the application of risk-neutral valuation, suppose that
the price of a non-dividend-Paying stock is $30 and consider a derivative that
pays off $100 in one year if the stock price is greater than $40 at that time.
(This is known as a binary cash-or-nothing call option.) Suppose that the riskfree rate (continuously compounded) is 5%, the expected return on the stock
(also continuously compounded) is 10%, and the stock price volatility is 30%
per annum. In a risk-neutral world the expected growth of the stock price is
5%. It can be shown (with the usual Black-Scholes lognormal assumptions)
that when the stock price has this growth rate the probability that the stock
price will be greater than $40 in one year is 0.1730. The expected payoff from
the derivatives is therefore 100 x 0.1730 = $17.30. The value of the derivative
is calculated by discounting this at 5%. It is $16.46.
The real-world (physical) probability of the stock price being greater than $40
in one year is calculated by assuming a growth rate of 10%. It is 0.2190. The
expected payoff in the real world is therefore $21.90. The problem with using
this expected cash flow is that we do not know the correct discount rate. The
stock price has risk associated with it that is priced by the market (otherwise the
expected return on the stock would not be 5% more than the risk-free rate). The
derivative has the effect of "leveraging this risk", so that a very high discount
rate is required for its expected payoff. Since we know the correct value of the
derivative is $16.46, we can deduce that the correct discount rate to apply to the
real-world expected payoff of $21.90 must be 28.6%.
Interestingly, the excess return on bonds varies through time. It increased steadily between 1997 and 2002 and then declined sharply in
2003 and 2004. For example, for the A-rated category the excess return
ranged from 35 basis points in 1997 to 119 basis points in 2002.
Real-World vs. Risk-Neutral Probabilities
The risk-neutral valuation argument is explained in Business Snapshot 11.1. It shows that we can value cash flows on the assumption that
all investors are risk neutral (i.e., on the assumption that they do not
require a premium for bearing risks). When we do this, we get the right
answer in the real world as well as in the risk-neutral world.
268
Chapter 11
The default probabilities implied from bond yields are risk-neutral
default probabilities (i.e., they are the probabilities of default in a world
where all investors are risk neutral). To understand why this is so, consider
the calculations of default probabilities in Table 11.3. These assume that
expected default losses can be discounted at the risk-free rate. The riskneutral valuation principle shows that this is a valid procedure provided the
expected losses are calculated in a risk-neutral world. This means that the
default probability Q in Table 11.3 must be a risk-neutral probability.
By contrast, the default probabilities implied from historical data are
real-world default probabilities (sometimes also called physical probabilities). The expected excess return in Table 11.5 arises directly from the
difference between real-world and risk-neutral default probabilities. If
there was no expected excess return, the real-world and risk-neutral
default probabilities would be the same, and vice versa.
Reasons for the Difference
Why do we see such big differences between real-world and risk-neutral
default probabilities? As we have just argued, this is the same as asking
why corporate bond traders earn more than the risk-free rate on average.
There are a number of potential reasons:
1. Corporate bonds are relatively illiquid and bond traders demand an
extra return to compensate for this. This may account for perhaps
25 basis points of the excess return. This is a significant part of the
excess return for high-quality bonds, but a relatively small part for
bonds rated Baa and below.
2. The subjective default probabilities of bond traders may be much
higher than the those given in Table 11.1. Bond traders may be
allowing for depression scenarios much worse than anything seen
during the 1970 to 2003 period. To test this, we can look at a table
produced by Moody's that is similar to Table 11.1, but applies to the
1920 to 2003 period instead of 1970 to 2003 period. When the
analysis is based on this table, historical default intensities for
investment-grade bonds in Table 11.4 rise somewhat. The Aaa
default intensity increases from 4 to 6 basis points; the Aa increases
from 6 to 22 basis points; the A increases from 13 to 29 basis points,
the Baa increases from 47 to 73 basis points. However, noninvestment-grade historical default intensities decline somewhat.
3. Bonds do not default independently of each other. This is the most
important reason for the results in Tables 11.4 and 11.5. There are
Credit Risk: Estimating Default Probabilities
269
periods of time when default rates are very low and periods of time
when they are very high. (Evidence for this can be obtained by
looking at the defaults rates in different years. Moody's statistics
show that between 1970 and 2003 the default rate per year ranged
from a low of 0.09% in 1979 to a high of 3.81% in 2001.) This gives
rise to systematic risk (i.e., risk that cannot be diversified away) and
bond traders should require an expected excess return for bearing the
risk. The variation in default rates from year to year may be because
of overall economic conditions or because a default by one company
has a ripple effect resulting in defaults by other companies. (The latter
is referred to by researchers as credit contagion.)
4. Bond returns are highly skewed with limited upside. As a result it is
much more difficult to diversify risks in a bond portfolio than in an
equity portfolio.11 A very large number of different bonds must be
held. In practice, many bond portfolios are far from fully diversified.
As a result bond traders may require an extra return for bearing
unsystematic risk as well as for bearing the systematic risk mentioned
in 3 above.
Which Estimates Should Be Used?
At this stage it is natural to ask whether we should use real-world or riskneutral default probabilities in the analysis of credit risk. The answer
depends on the purpose of the analysis. When valuing credit derivatives
or estimating the impact of default risk on the pricing of instruments, we
should use risk-neutral default probabilities. This is because the analysis
calculates the present value of expected future cash flows and almost
invariably (implicitly or explicitly) involves using risk-neutral valuation.
When carrying out scenario analyses to calculate potential future losses
from defaults, we should use real-world default probabilities. The PD
used to calculate regulatory capital is a real-world default probability.
11.6 USING EQUITY PRICES TO ESTIMATE DEFAULT
PROBABILITIES
When we use a table such as Table 11.1 to estimate a company's realworld probability of default, we are relying on the company's credit
11
See J. D. Amato and E. M. Remolona, "The Credit Spread Puzzle," BIS Quarterly
Review, 5 (December 2003), 51-63.
270
Chapter 11
rating. Unfortunately, credit ratings are revised relatively infrequently.
This has led some analysts to argue that equity prices can provide more
up-to-date information for estimating default probabilities.
In 1974, Merton proposed a model where a company's equity is an
option on the assets of the company. 12 Suppose, for simplicity, that a firm
has one zero-coupon bond outstanding and that the bond matures at
time T. Define:
Value of company's assets today
Value of company's assets at time T
Value of company's equity today
Value of company's equity at time T
Amount of debt interest and principal due to be repaid at time T
Volatility of assets (assumed constant)
Instantaneous volatility of equity
If VT < D, it is (at least in theory) rational for the company to default on
the debt at time T. The value of the equity is then zero. If VT > D, the
company should make the debt repayment at time T and the value of the
equity at this time is VT — D. Merton's model, therefore, gives the value of
the firm's equity at time T as
ET = max(VT - D, 0)
This shows that the equity of a company is a call option on the value of
the assets of the company with a strike price equal to the repayment
required on the debt. The Black-Scholes formula (see Appendix C at the
end of this book) gives the value of the equity today as
where
and N is the cumulative normal distribution function. The value of the
debt today is V0 — E0.
The risk-neutral probability that the company will default on the debt
is N(-d2). To calculate this, we require V0 and
Neither of these are
12
See R. Merton "On the Pricing of Corporate Debt: The Risk Structure of Interest
Rates." Journal of Finance, 29 (1974), 449-470 .
Credit Risk: Estimating Default Probabilities
271
directly observable. However, if the company is publicly traded, we can
observe E0. This means that equation (11.4) provides one condition that
must be satisfied by V0 and
We can also estimate
From a result in
stochastic calculus known as
lemma, we have
or
This provides another equation that must be satisfied by V0 and
Equations (11.4) and (11.5) provide a pair of simultaneous equations
13
that can be solved for V0 and
Example 11.2
The value of a company's equity is $3 million and the volatility of the equity is
80%. The debt that will have to be paid in one year is $10 million. The riskfree rate is 5% per annum. In this case, E0 — 3,
= 0.80, r = 0.05, T = 1,
and D = 1 0 . Solving equations (11.4) and (11.5) yields V0 = 12.40 and
= 0.2123. The parameter d2 is 1.1408, so that the probability of default
is N(-d2) = 0.127, or 12.7%. The market value of the debt is V0 - E0, or 9.40.
The present value of the promised payment on the debt is 10e -0.05xl = 9.51.
The expected loss on the debt is therefore (9.51 — 9.40)/9.51, or about 1.2% of
its no-default value. Comparing this with the probability of default gives the
expected recovery in the event of a default as (12.7 — 1.2)/12.7, or about 91%.
Distance to Default
Moody's KMV has coined the term distance to default to describe the
output from Merton's model. This is the number of standard deviations
by which the asset price must change for default to be triggered T years in
the future. It is given by
As the distance to default decreases, the company becomes more likely to
default. In Example 11.2 the one-year distance to default is 1.14 standard
deviations.
13
To solve two nonlinear equations of the form F(x, y) = 0 and G(x, y) = 0, we can use
the Solver routine in Excel to find the values of x and y that minimize [F(x, y)]2+
272
Chapter 11
Extensions of the Basic Model
The basic Merton model we have just presented has been extended in a
number of ways. For example, one version of the model assumes that a
default occurs whenever the value of the assets falls below a barrier level.
Another allows payments on debt instruments to be required at more than
one time. Many analysts have found the implied volatility of equity issued
by a company to be a good predictor of the probability of default. (The
higher the implied volatility, the higher the probability of default.) Hull et
al. show that this is consistent with Merton's model.14 They provide a way
of implementing Merton's model using two equity implied volatilities and
show that the resulting model provides results comparable to those
provided by the usual implementation of the model.
Performance of the Model
How well do the default probabilities produced by Merton's model and
its extensions correspond to actual default experience? The answer is that
Merton's model and its extensions produce a good ranking of default
probabilities (risk neutral or real world). This means that a monotonic
transformation can be estimated to convert the probability of default
output from Merton's model into a good estimate of either the real-world
or risk-neutral default probability.15 It may seem strange to use a default
probability, N(—d2), that is in theory a risk-neutral default probability to
estimate a real-world default probability. Given the nature of the calibration process we have just described, the underlying assumption is that the
ranking of risk-neutral default probabilities of different companies is the
same as that of their real-world default probabilities.
SUMMARY
The probability that a company will default during a particular period 0f
time in the future can be estimated from historical data, bond prices, or
equity prices. The default probabilities calculated from bond prices are
risk-neutral probabilities, whereas those calculated from historical data
14
See J. Hull, I. Nelken, and A. White, "Merton's Model, Credit Risk, and Volatility
Skews," Journal of Credit Risk, 1, No. 1 (2004), 1-27.
15
Moody's KMV provides a service that transforms a default probability produced by
Merton's model into a real-world default probability (which it refers to as an EDF, short
for expected default frequency). CreditGrades uses Merton's model to estimate credit
spreads, which are closely linked to risk-neutral default probabilities.
Credit Risk: Estimating Default Probabilities
273
are real-world probabilities. The default probabilities calculated from
equity prices using Merton's model are in theory risk-neutral default
probabilities. However, the output from the model can be calibrated so
that either risk-neutral or real-world default probabilities are produced.
Real-world probabilities should be used for scenario analysis and the
calculation of credit VaR. Risk-neutral probabilities should be used for
valuing credit-sensitive instruments. Risk-neutral default probabilities are
usually significantly higher than real-world probabilities.
FURTHER READING
Altman, E. I., "Measuring Corporate Bond Mortality and Performance,"
Journal of Finance, 44 (1989): 902-922.
Duffie, D., and K. Singleton, "Modeling Term Structures of Defaultable
Bonds," Review of Financial Studies, 12 (1999): 687-720.
Hull, J., M. Predescu, and A. White, "Relationship between Credit Default
Swap Spreads, Bond Yields, and Credit Rating Announcements," Journal of
Banking and Finance, 28 (November 2004): 2789-2811.
Hull, J., M. Predescu, and A. White, "Bond Prices, Default Probabilities, and
Risk Premiums," Journal of Credit Risk, 1, No. 2 (2004): 53-60.
Kealhofer, S., "Quantifying Default Risk I: Default Prediction," Financial
Analysts Journal, 59, No. 1 (2003): 30-44.
Kealhofer, S., "Quantifying Default Risk II: Debt Valuation," Financial
Analysts Journal, 59, No. 3 (2003), 78-92.
Litterman, R., and T. Iben, "Corporate Bond Valuation and the Term Structure
of Credit Spreads," Journal of Portfolio Management, Spring 1991: 52-64.
Merton, R. C, "On the Pricing of Corporate Debt: The Risk Structure of
Interest Rates," Journal of Finance, 29 (1974): 449-470.
Rodriguez, R. J., "Default Risk, Yield Spreads, and Time to Maturity," Journal
of Financial and Quantitative Analysis, 23 (1988): 111-117.
QUESTIONS A N D PROBLEMS (Answers at End of Book)
11.1. How many ratings does Moody's use for companies that have not
defaulted? What are they?
11.2. How many ratings does S&P use for companies that have not defaulted?
What are they?
11.3. Calculate the average default intensity for B-rated companies during the
first year from the data in Table 11.1.
274
Chapter \\
11.4. Calculate the average default intensity for Ba-rated companies during the
third year from the data in Table 11.1.
11.5. The spread between the yield on a 3-year corporate bond and the yield on
a similar risk-free bond is 50 basis points. The recovery rate is 30%,
Estimate the average default intensity per year over the 3-year period.
11.6. The spread between the yield on a 5-year bond issued by a company and
the yield on a similar risk-free bond is 80 basis points. Assume a recovery
rate of 40%. Estimate the average default intensity per year over the
5-year period. If the spread is 70 basis points for a 3-year bond, what do
your results indicate about the average default intensity in years 4 and 5?
11.7. Should researchers use real-world or risk-neutral default probabilities for
(a) calculating credit value at risk and (b) adjusting the price of a
derivative for defaults?
11.8. How are recovery rates usually defined?
11.9. Verify (a) that the numbers in the second column of Table 11.4 are
consistent with the numbers in Table 11.1 and (b) that the numbers in
the fourth column of Table 11.5 are consistent with the numbers in
Table 11.4 and a recovery rate of 40%.
11.10. A 4-year corporate bond provides a coupon of 4% per year payable
semiannually and has a yield of 5% expressed with continuous compounding. The risk-free yield curve is flat at 3% with continuous compounding.
Assume that defaults can take place at the end of each year (immediately
before a coupon or principal payment) and the recovery rate is 30%.
Estimate the risk-neutral default probability on the assumption that it is
the same each year.
11.11. A company has issued 3- and 5-year bonds with a coupon of 4% per
annum payable annually. The yields on the bonds (expressed with
continuous compounding) are 4.5% and 4.75%, respectively. Risk-free
rates are 3.5% with continuous compounding for all maturities. The
recovery rate is 40%. Defaults can take place halfway through each year.
The risk-neutral default rates per year are Q1 for years 1 to 3 and Q2 for
years 4 and 5. Estimate Q1 and Q2.
11.12. Suppose that in an asset swap B is the market price of the bond per dollar
of principal, B* is the default-free value of the bond per dollar 01
principal, and V is the present value of the asset swap spread per dollar
of principal. Show that V = B* — B.
11.13. Show that under Merton's model in Section 11.6 the credit spread on a
T-year zero-coupon bond is
-ln[N{d2)
where L = De-rT/V0.
+ N(-d1)/L]/T
Credit Risk:
Estimating Default Probabilities
275
11.14. The value of a company's equity is $2 million and the volatility of its
equity is 50%. The debt that will have to be repaid in one year is
$5 million. The risk-free interest rate is 4% per annum. Use Merton's
model to estimate the probability of default. {Hint: The Solver function
in Excel can be used for this question.)
11.15. Suppose that the LIBOR/swap curve is flat at 6% with continuous
compounding and a 5-year bond with a coupon of 5% (paid semiannually) sells for 90.00. How would an asset swap on the bond be
structured? What is the asset swap spread that would be calculated in
this situation?
ASSIGNMENT QUESTIONS
11.16. Suppose a 3-year corporate bond provides a coupon of 7% per year
payable semiannually and has a yield of 5% (expressed with semiannual
compounding). The yields for all maturities on risk-free bonds is 4% per
annum (expressed with semiannual compounding). Assume that defaults
can take place every 6 months (immediately before a coupon payment)
and the recovery rate is 45%. Estimate the default probabilities, assuming
(a) that the unconditional default probabilities are the same on each
possible default date and (b) that the default probabilities conditional on
no earlier default are the same on each possible default date.
11.17. A company has 1- and 2-year bonds outstanding, each providing a
coupon of 8% per year payable annually. The yields on the bonds
(expressed with continuous compounding) are 6.0% and 6.6%, respectively. Risk-free rates are 4.5% for all maturities. The recovery rate is
35%. Defaults can take place halfway through each year. Estimate the
risk-neutral default rate each year.
11.18. The value of a company's equity is $4 million and the volatility of its
equity is 60%. The debt that will have to be repaid in 2 years is
$15 million. The risk-free interest rate is 6% per annum. Use Merton's
model to estimate the expected loss from default, the probability of
default, and the recovery rate in the event of default. Explain why
Merton's model gives a high recovery rate. {Hint: The Solver function
in Excel can be used for this question.)
Credit Risk Losses
and Credit VaR
This chapter starts by discussing the nature of the credit risk in derivatives transactions. It shows how expected credit losses can be calculated
and looks at the various ways in which a financial institution can
structure its contracts so as to reduce these expected credit losses.
The last part of the chapter covers the calculation of credit VaR. The
1986 amendment to Basel I allows banks to develop their own models for
calculating VaR for market risk in the trading book. However, Basel II
does not give banks quite the same freedom to calculate credit VaR for
the banking book. The development of internal models for calculating
credit VaR is nevertheless an important activity for banks. These models
can be used to determine regulatory capital for credit risk in the trading
book (the specific risk capital charge). They can also be used when
economic capital is calculated, as we will explain in Chapter 16.
Chapter 11 covered the important difference between real-world and
risk-neutral default probabilities (see Section 11.5). Real-world default
Probabilities can be estimated from historical data. Risk-neutral default
Probabilities can be estimated from bond prices. In the first part of this
chapter, when calculating expected credit losses, we will use risk-neutral
default probabilities. This is because we are calculating the present value
of future cash flows. Later in the chapter, when calculating credit VaR,
we will use real-world default probabilities. This is because we are looking
at future scenarios and not calculating present values.
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Chapter 12
12.1 ESTIMATING CREDIT LOSSES
Credit losses on a loan depend primarily on the probability of default and
the recovery rate. We covered estimates of the probability of default in
Chapter 11. The estimate of recovery rate (or equivalently loss given
default) depends on the nature of the collateral, if any. When making
estimates, a financial institution is likely to use a mixture of its own
experience and statistics published by rating agencies such as those in
Table 11.2.
Derivatives Transactions
The credit exposure on a derivatives transaction is more complicated than
that on a loan. This is because the claim that will be made in the event of
a default is more uncertain than in the case of a loan. Consider a financial
institution that has one derivatives contract outstanding with a counterparty. We can distinguish three possible situations:
1. The contract is always a liability to the financial institution.
2. The contract is always an asset to the financial institution.
3. The contract can become either an asset or a liability to the financial
institution.
An example of a derivatives contract in the first category is a short option
position; an example in the second category is a long option position; an
example in the third category is a forward contract or a swap.
Derivatives in the first category have no credit risk to the financial
institution. If the counterparty goes bankrupt, there will be no loss. The
derivative is one of the counterparty's assets. It is likely to be retained,
closed out, or sold to a third party. The result is no loss (or gain) to the
financial institution.
Derivatives in the second category always have credit risk to the
financial institution. If the counterparty goes bankrupt, a loss is likely
to be experienced. The derivative is one of the counterparty's liabilities.
The financial institution has to make a claim against the assets of the
counterparty and may eventually realize some percentage of the value of
the derivative.1
Derivatives in the third category may or may not have credit risk. If
the counterparty defaults when the value of the derivative is positive to
1
Counterparties to derivatives transactions usually rank equally with unsecure
creditors in the event of a liquidation.
Credit Risk Losses and Credit VaR
279
the financial institution, then a claim will be made against the assets of
the counterparty and a loss is likely to be experienced. If the counterparty defaults when the value is negative to the financial institution, no
loss is made because the derivative will be retained, closed out, or sold to
a third party. 2
Adjusting Derivatives Valuations for Counterparty Default Risk
How should a financial institution (or end-user of derivatives) adjust the
value of a derivative to allow for counterparty credit risk? Consider a
derivative that has a value of f0 today assuming no defaults. Let us
suppose that defaults can take place at times t1, t2,..., tn and that the
value of the derivative to the financial institution (assuming no defaults)
at time ti is fi. Define the risk-neutral probability of default at time ti as
qi and the expected recovery rate as R.3
The exposure at time ti is the financial institution's potential loss. This is
max(fi, 0) as illustrated in Figure 12.1. Assume that the expected recovery
in the event of a default is R times the exposure. Assume also that the
recovery rate and the probability of default are independent of the value of
the derivative. The risk-neutral expected loss from default at time ti is
where
denotes expected value in a risk-neutral world. Taking present
values leads to the cost of defaults being
where ui equals qi{1 — R) and vi is the value today of an instrument that
Pays off the exposure on the derivative under consideration at time ti.
Consider again the three categories of derivatives mentioned earlier.
The first category (where the derivative is always a liability to the financial
institution) is easy to deal with. The value of fi is always negative, and so
the total expected loss from defaults given by equation (12.1) is always
zero. The financial institution needs to make no adjustments for the cost
2
A company usually defaults because of the total amount of its liabilities, not because of
the value of any one transaction. At the time a company defaults it is likely that some of
its contracts will have positive values.
3
This probability of default can be calculated from bond prices in the way described in
Section 11.4.
Chapter 12
280
Figure 12.1
Exposure on a derivative as a function of
its no-default value.
of defaults. (Of course, the counterparty may want to take account of the
possibility of the financial institution defaulting in its own analysis.)
For the second category (where the derivative is always an asset to the
financial institution), the value of fi is always positive. The expression
max(fi, 0) is always equal to fi. Since vi is the present value of fi, it
always equals f 0 . 4 The expected loss from default is therefore f0 times the
total probability of default during the life of the derivative times 1 — R.
Example 12.1
Consider a two-year over-the-counter option with a value (assuming no
defaults) of $3. Suppose that the company selling the option has a riskneutral probability of defaulting during the two-year period of 4% and the
recovery in the event of a default is 25%. The expected cost of defaults is
3 x 0.04 x (1 - 0.25), or $0.09. The buyer of the option should therefore be
prepared to pay only $2.91.
For the third category of derivatives, the sign of fi is uncertain. The
variable vi is a call option on fi with a strike price of zero. One way of
calculating all the vi with a single analysis is to simulate the underlying
market variables over the life of the derivative. Sometimes approximate
analytic calculations are possible (see Problems 12.7, 12.8, and 12.14).
4
This assumes no payoffs from the derivative prior to time ti. The analysis can be
adjusted to cope with situations where there are intermediate payoffs.
Credit Risk Losses and Credit VaR
281
Example 12.2
Consider an interest rate swap where a bank is receiving fixed and paying
floating. The exposure on the swap at a particular future time ti is
maxfV(ti), 0]
where V(ti) is the value of the swap at time ti. This is the payoff from a swap
option. The variable vi is therefore the value today of a type of European swap
option exercisable at time ti.
The analyses we have presented assume that the probability of default is
independent of the value of the derivative. This is likely to be a reasonable
approximation in circumstances when the derivative is a small part of the
portfolio of the counterparty or when the counterparty is using the
derivative for hedging purposes. When a counterparty wants to enter into
a large derivatives transaction for speculative purposes, a financial institution should be wary. When the transaction has a large negative value for
the counterparty (and a large positive value for the financial institution),
the probability of the counterparty declaring bankruptcy may be much
higher than when the situation is the other way round.
Interest Rate Swaps vs. Currency Swaps
The impact of default risk on interest rate swaps is considerably less than
that on currency swaps. Figure 12.2 shows the reason for this. It compares
the expected exposure for a bank on a pair of offsetting interest rate swaps
Figure 12.2 Expected exposure on a matched pair of interest
rate swaps and a matched pair of currency swaps.
282
Chapter 12
with different counterparties to the expected exposure on a pair of
offsetting currency swaps with different counterparties. The expected
exposure on the interest rate swaps starts at zero, increases, and then
decreases to zero. By contrast, expected exposure on a the currency swaps
increases steadily with the passage of time.5 The reason for the difference
is largely that principals are exchanged at the end of the life of a currency
swap and there is uncertainty about the exchange rate at that time.6
Two-Sided Default Risk
One of the interesting aspects of contracts in the third category (i.e.,
contracts that can become assets or liabilities) is that there is two-sided
default risk. For example, if Company X enters into a currency swap with
Company Y, Company X may lose money if Company Y defaults and
vice versa. Human nature being what it is, most financial and nonfinancial companies consider that there is no chance that they themselves will
default but want to make an adjustment to contract terms for a possible
default by their counterparty. In our example, Company X wants to be
compensated for the possibility that Company Y will default and Company Y wants to be compensated for the possibility that Company X will
default. This can make it very difficult for the two companies to agree on
terms and explains why it is difficult for financial institutions that are not
highly creditworthy to be active in the derivatives market.
In theory, the value of the currency swap to Company X should be
f-y+
x
where f is the value if both sides were default-free, y is the adjustment
necessary for the possibility of Company Y defaulting, and x is the
adjustment necessary for Company X itself defaulting. Similarly, the value
to Company Y should be
-f - x + y
In practice, Company X is likely to calculate a value of f — y and
Company Y is likely to calculate a value of — f — x. Unless x and y are
small, it is unlikely that the companies will be able to agree on a price.
Continuing with our currency swap example, suppose that interest rates
in the foreign and domestic currency are the same and that Company X
5
The expected exposure affects the vi in equation (12.1). The ui are the same for both
types of swaps.
6
Currency swaps with no final exchange of principal have become more common in
recent years. These have less credit risk and market risk than traditional currency swaps.
Credit Risk Losses and Credit VaR
283
and Company Y are equally creditworthy. The swap is symmetrical in the
sense that it has just as much chance of moving in the money for
Company X as for Company Y. In this case, x = y and neither side
should in theory make any adjustment for credit risk.
12.2 CREDIT RISK M I T I G A T I O N
In many cases the analysis we have just presented overstates the credit risk
in a derivatives transaction. This is because there are a number of clauses
that derivatives dealers include in their contracts to mitigate credit risk.
Netting
We discussed netting in Section 7.5. A netting clause in a derivatives
contract states that if a company defaults on one contract it has with a
counterparty then it must default on all outstanding contracts with the
counterparty. Netting has been successfully tested in the courts in most
jurisdictions and can substantially reduce credit risk for a financial
institution. We can extend the analysis presented in the previous section
so that equation (12.1) gives the present value of the expected loss from
all contracts with a counterparty when netting agreements are in place.
This is achieved by redefining vi in the equation as the present value of a
derivative that pays off the exposure at time ti on the portfolio of all
contracts with a counterparty.
A challenging task for a financial institution when considering whether
it should enter into a new derivatives contract with a counterparty is to
calculate the incremental effect on expected credit losses. This can be
done by using equation (12.1) in the way just described to calculate
expected default costs with and without the contract. It is interesting to
note that, because of netting, the incremental effect of a new contract on
expected default losses can be negative. This tends to happen when the
value of the new contract is highly negatively correlated with the value of
eisting contracts.
It might be thought that in well-functioning capital markets a company
wanting to enter into a derivatives transaction will get the same quote
from all dealers. Netting shows that this is not necessarily the case. The
company is likely to get the most favorable quote from a financial
institution it has done business with in the past—particularly if that
business gives rise to exposures for the financial institution that are
opposite to the exposure generated by the new transaction.
284
Chapter 12
Collateralization
Another clause frequently used to mitigate credit risks is known as
Collateralization. Suppose that a company and a financial institution have
entered into a number of derivatives contracts. A typical Collateralization
agreement specifies that the contracts be marked to market periodically
using a pre-agreed formula. If the total value of the contracts to the
financial institution is above a certain threshold level on a certain day, it
can ask the company to post collateral. The amount of collateral posted,
when added to collateral already posted, by the company is equal to the
difference between the value of the contracts to the financial institution
and the threshold level. When the contracts move in favor of the company, so that the difference between value of the contracts to the financial
institution and the threshold level is less than the total margin already
posted, the company can reclaim margin. In the event of a default by the
company, the financial institution can seize the collateral. If the company
does not post collateral as required, the financial institution can close out
the contracts. Long-Term Capital Management made extensive use of
Collateralization agreements (see Business Snapshot 12.1).
Suppose, for example, that the threshold level for the company is
$10 million and contract is marked to market daily for the purposes of
Collateralization. If on a particular day the value of the contract to the
financial institution is $10.5 million, it can ask for $0.5 million of collateral. If on the next day the value of the contract rises to $11.4 million, it
can ask for a further $0.9 million of collateral. If the value of the contract
falls to $10.8 million on the following day, the company can ask for
$0.6 million of the collateral to be returned. Note that the threshold
($10 million in this case) can be regarded as a line of credit that the
financial institution is prepared to grant to the company.
Collateral must be deposited by the company with the financial institution in cash or in the form of acceptable securities such as bonds. Interest
is normally paid on cash. The securities are subject to a discount known
as a haircut applied to their market value for the purposes of collateral
calculations.
If the Collateralization agreement is a two-way agreement, a threshold
will also be specified for the financial institution. The company can then
ask the financial institution to post collateral when the marked-to-market
value of the outstanding contracts to the company exceeds the threshold.
Collateralization agreements provide a great deal of protection against
the possibility of default (just as the margin accounts discussed in
Credit Risk Losses and Credit VaR
Business Snapshot 12.1
285
Long-Term Capital Management's Big Loss
Long-Term Capital Management (LTCM), a hedge fund formed in the mid1990s, always collateralized its transactions. The hedge fund's investment
strategy was known as convergence arbitrage. A very simple example of what
it might do is the following. It would find two bonds, X and Y, issued by the
same company, promising the same payoffs, with X being less liquid (i.e., less
actively traded) than Y. The market always places a value on liquidity. As a
result, the price of X would be less than the price of Y. LTCM would buy X and
short Y, and wait, expecting the prices of the two bonds to converge at some
future time.
When interest rates increased, the company expected both bonds to move
down in price by about the same amount, so that the collateral it paid on bond
X would be about the same as that which it received on bond Y. Similarly, when
interest rates decreased, LTCM expected both bonds to move up in price by
about the same amount, so that the collateral it received on bond X would be
about the same as that which it paid on bond Y. It therefore expected no
significant outflow of funds as a result of its Collateralization agreements.
In August 1998, Russia defaulted on its debt and this led to what is termed a
"flight to quality" in capital markets. One result was that investors valued
liquid instruments more highly than usual and the spreads between the prices
of the liquid and illiquid instruments in LTCM's portfolio increased dramatically. The prices of the bonds LTCM had bought went down and the prices of
those it had shorted increased. It was required to post collateral on both. The
company was highly leveraged and unable to make the payments required
under the Collateralization agreements. The result was that positions had to be
closed out and there was a total loss of about $4 billion. If the company had
been less highly leveraged, it would probably have been able to survive the
flight to quality and could have waited for the prices of the liquid and illiquid
bonds to become closer.
Chapter 2 provide protection for people who trade on an exchange).
However, the threshold amount is not subject to protection. Furthermore,
even when the threshold is zero, the protection is not total. When a
company gets into financial difficulties it is likely to stop responding to
requests to post collateral. By the time the counterparty exercises its right
to close out contracts, their value may have moved further in its favor.
Downgrade Triggers
Another credit mitigation technique used by a financial institution is
known as a downgrade trigger. This is a clause stating that if the credit
286
Business Snapshot 12.2
Chapter 12
Downgrade Triggers and Enron's Bankruptcy
In December 2001, Enron, one of the largest companies in the United States,
went bankrupt. Right up to the last few days, it had an investment-grade credit
rating. The Moody's rating immediately prior to default was Baa3 and the
S&P rating was BBB—. The default was, however, anticipated to some extent
by the stock market because Enron's stock price fell sharply in the period
leading up to the bankruptcy. The probability of default estimated by models
such as the one described in Section 11.6 increased sharply during this period.
Enron had entered into a huge number of derivatives contracts with downgrade triggers. The downgrade triggers stated that, if its credit rating fell below
investment grade (i.e., below Baa3/BBB—), its counterparties would have the
option of closing out contracts. Suppose that Enron had been downgraded to
below investment grade in, say, October 2001. The contracts that counterparties would choose to close out would be those with a negative values to
Enron (and positive values to the counterparties). As a result Enron would
have been required to make huge cash payments to its counterparties. It would
not have been able to do this and immediate bankruptcy would result.
This example illustrates that downgrade triggers provide protection only
when relatively little use is made of them. When a company enters into a huge
number of contracts with downgrade triggers, they may actual cause a company to go bankrupt prematurely. In Enron's case we could argue that it was
going to go bankrupt anyway and accelerating the event by two months would
not have done any harm. In fact, Enron did have a chance of survival in
October 2001. Attempts were being made to work out a deal with another
energy company, Dynergy, and so forcing bankruptcy in October 2001 was
not in the interests of either creditors or shareholders.
The credit rating companies found themselves in a difficult position. If they
downgraded Enron to recognize its deteriorating financial position, they were
signing its death warrant. If they did not do so, there was a chance of Enron
surviving.
rating of the counterparty falls below a certain level, say Baa, then the
financial institution has the option to close out a derivatives contract at its
market value. (A procedure for determining the value of the contract
must be agreed to in advance.)
Downgrade triggers do not provide protection from a big jump in a
company's credit rating (e.g., from A to default). Also, downgrade
triggers work well only when relatively little use is made of them. If a
company has entered into downgrade triggers with many counterparties'
they are liable to provide relatively little protection to the counterparties
(see Business Snapshot 12.2).
Credit Risk Losses and Credit VaR
287
12.3 CREDIT VaR
Credit value at risk can be defined in the same way as we defined market
value at risk in Chapter 8. For example, a credit VaR with a confidence
level of 99.9% and a one-year time horizon is the credit loss that we are
99.9% confident will not be exceeded over one year. Whereas the time
horizon for market risk is usually between one day and one month that
for credit risk is usually much longer—often one year.
For regulatory purposes banks using the internal ratings based
approach must calculate credit VaR for items in the banking book using
the methodology prescribed by the Basel Committee. Banks are given
freedom in making their own estimates of probability of default, PD. 7
However, they must use the correlation model and correlation parameters
specified by the Basel Committee.
When calculating credit VaR for specific risk, banks have more freedom. Specific risk is the risk in the trading book that is related to the
credit quality of individual companies. Although standard rules for
determining specific risk have been specified, banks can, with regulatory
approval, use their own models. For a model to be approved, the bank
supervisor must be satisfied that concentration risk, spread risk, downgrade risk, and default risk are appropriately captured. The capital charge
for specific risk is the product of a multiplier and the ten-day 99% VaR,
with the minimum level for the multiplier being 4.
12.4 VASICEK'S MODEL
Vasicek's model, which we discussed in Sections 6.5 and 7.8, provides an
easy way to estimate credit VaR for a loan portfolio. From
equation (6.12), there is a probability X that the percentage of defaults
on a large portfolio by time T is, in a one-factor Gaussian copula model,
less than
where Q(T) is the cumulative probability of each loan defaulting by time T
and is the copula correlation. When multiplied by the average exposure
per loan and by the average loss given default, this gives the T-year VaR for
7
As mentioned earlier, the probability of default in credit VaR calculations should be a
real-world probability of default, not a risk-neutral probability of default.
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Chapter 12
an X% confidence level. As explained in Chapter 7, the Basel Committee
has based the capital it requires for credit risk in the banking book on this
model. The time horizon T is one year, the confidence level X is 99.9%
and the value of is determined by the Basel Committee. As we saw in
Chapter 7, in some cases depends on the one-year probability of default.
12.5 CREDIT RISK PLUS
In 1998 Credit Suisse Financial Products proposed a methodology for
calculating VaR that it termed Credit Risk Plus.8 It utilizes ideas that are
well established in the insurance industry. We will present a simplified
version of the approach.
Suppose that a financial institution has N counterparties of a certain
type and the probability of default by each counterparty in time T is p.
The expected number of defaults,
for the whole portfolio is given by
Assuming that default events are independent and that p is
small, the probability of n defaults is given by the Poisson distribution as
This can be combined with a probability distribution for the losses
experienced on a single counterparty default (taking account of the
impact of netting) to obtain a probability distribution for the total default
losses from the counterparties. To estimate the probability distribution
for losses from a single counterparty default, we can look at the current
probability distribution of our exposures to counterparties and adjust this
according to historical recovery rates.
In practice, it is likely to be necessary for the financial institution to
consider several categories of counterparties. This means that the analysis
just described must be done for each category and the results combined.
Another complication is that default rates vary significantly from year to
year. Data provided by Moody's show that the default rate per year for all
bonds during the 1970 to 1999 period ranged from 0.09% in 1979 to 3.52%
in 2001. To account for this, we can assume a probability distribution for
the overall default rate based on historical data such as that provided by
8
See Credit Suisse Financial Products, "Credit Risk Management Framework,
October, 1997.
Credit Risk Losses and Credit VaR
Figure 12.3
289
Probability distribution of default losses.
Moody's. The probability of default for each category of counterparty can
then be assumed to be linearly dependent on the overall default rate.
Credit Suisse Financial Products show that, if certain assumptions are
made, the total loss probability distribution can be calculated analytically.
To accommodate more general assumptions, Monte Carlo simulation can
be used. The procedure is as follows:
1. Sample an overall default rate.
2. Calculate a probability of default for each category.
3. Sample a number of defaults for each category.
4. Sample a loss for each default.
5. Calculate total loss.
6. Repeat Steps 1 to 5 many times.
The effect of assuming a probability distribution for default rates in the
way just described is to build in default correlations. This makes the
Probability distribution of total default losses positively skewed, as
indicated in Figure 12.3.
12-6 CREDITMETRICS
Credit Risk Plus estimates the probability distribution of losses arising
from defaults. As indicated earlier, regulators like the internal models
used for specific risk to reflect losses from downgrades and credit spread
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Chapter 12
changes as well as defaults. This has led many banks to implement
CreditMetrics for specific risk calculations.
CreditMetrics was proposed by J.P. Morgan in 1997. It is based on an
analysis of credit migration. This is the probability of a company moving
from one rating category to another during a certain period of time. A
typical one-year ratings transition matrix is shown in Table 12.1. \\
indicates that the percentage probability of a bond moving from one
rating category to another during a one-year period. For example, a bond
that starts with an A credit rating has a 91.83% chance of still having an
A rating at the end of one year. It has a 0.02% chance of defaulting
during the year, a 0.13% chance of dropping to B, and so on.
Consider a bank with a portfolio of corporate bonds. Calculating a oneyear VaR for the portfolio using CreditMetrics involves carrying out
Monte Carlo simulation of ratings transitions for bonds in the portfolio
over a one-year period. On each simulation trial the final credit rating of all
bonds is calculated and the bonds are revalued to determine total credit
losses for the year. The 99% worst result is the one-year 99% VaR for the
portfolio.
It is interesting to note that if both the CreditMetrics and the Credit
Risk Plus models are accurate, they should predict the same probability
distribution for losses over the long term. It is the timing of losses that is
different. Suppose, for example, that you hold a certain bond in your
portfolio. In year 1 it gets downgraded from A to BBB; in year 2 it gets
downgraded from BBB to B; in year 3 it defaults. You could assume that
there are no losses in years 1 and 2 and calculate the loss in year 3 (the
Credit Risk Plus approach). Alternatively, you can calculate separate
Table 12.1 One-year ratings transition matrix (probabilities expressed as
percentages). From results reported by Moody's in 2004.
Initial
rating
Rating at year-end
Aaa
Aa
A
Baa
Ba
B
Caa
Default
Aaa
Aa
A
Baa
Ba
B
Caa
Default
92.18
1.17
0.05
0.05
0.01
0.01
0.00
0.00
7.06
90.85
2.39
0.24
0.05
0.03
0.00
0.00
0.73
7.63
91.83
5.20
0.50
0.13
0.00
0.00
0.00
0.26
5.07
88.48
5.45
0.43
0.58
0.00
0.02
0.07
0.50
4.88
85.13
6.52
1.74
0.00
0.00
0.01
0.13
0.80
7.05
83.20
4.18
0.00
0.00
0.00
0.01
0.16
0.55
3.04
67.99
0.00
0.00
0.02
0.02
0.18
1.2?
6.64
25.50
100.00
Credit Risk Losses and Credit VaR
291
revaluation losses in years 1, 2, and 3 (the CreditMetrics approach). The
losses under the second approach should in theory add up to the losses
under the first approach.
There are two key issues in the implementation of CreditMetrics. One is
handling correlations between bonds. The other is calculating credit
spreads for valuing the bonds at the end of the year.
The CreditMetrics Correlation Model
In sampling to determine credit losses, the credit rating changes for
different counterparties should not be assumed to be independent. A
Gaussian copula model can be used to construct a joint probability
distribution of rating changes (see Section 6.4 for a discussion of copula
models). The copula correlation between the rating transitions for two
companies is typically set equal to the correlation between their equity
returns using a factor model similar to that in Section 6.3.
As an illustration of the CreditMetrics approach, suppose that we are
simulating the rating change of an A-rated and a B-rated company over a
one-year period using the transition matrix in Table 12.1. Suppose that
the correlation between the equity returns of the two companies is 0.2. On
each simulation trial we would sample two variables xA and xB from
standard normal distributions, so that their correlation is 0.2. The
variable xA determines the new rating of the A-rated company and
variable xB determines the new rating of the B-rated company. Since
the A-rated company gets upgraded to Aaa if xA < —3.2905, it becomes
Aa if -3.2905 <xA< -1.9703, it stays A if -1.9703 <xA< 1.5779, and
so on. Similarly, since
the B-rated company becomes Aaa if xB < —3.7190, it becomes Aa if
-3.7190 < xB < -3.3528, it becomes A if -3.3528 < xB < -2.9290, and
so on. The A-rated company defaults if xA >
that is, when
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Chapter 12
Figure 12.4 The CreditMetrics correlation model: transition of an A-rated and
a B-rated company to a new rating after one year. Here xA and xB are sampled
from standard normal distributions with the correlation equal to the correlation
between the equity returns of A and B.
xA > 3.5401. The B-rated company defaults when xB > N -1 (0.9336), that
is, when xB > 1.5032. This example is illustrated in Figure 12.4.
Spread Changes
In order to revalue the portfolio of bonds on each simulation trial, it is
necessary to calculate spread changes. One way of proceeding is to use a
one-factor regression model to divide the spread changes for each bond
into a systematic component and an idiosyncratic component. The systematic component is the spread change that affects all bonds in the
rating category being considered. The idiosyncratic component affects
only one particular bond.
If the systematic component of spread changes is reflected in market risk
VaR calculations, it is only necessary to take account of the idiosyncratic
Credit Risk Losses and Credit VaR
293
spread changes in specific risk VaR calculations. On each trial, therefore,
we calculate not only what happens to the rating of each bond but also an
idiosyncratic spread change for the bond. For each bond on each simulation trial, one of three things can happen:
1. The credit rating of the bond stays the same. In this case the bond is
revalued in a way that reflects the idiosyncratic spread change for the
bond.
2. The credit rating of the bond changes. In this case the bond is
revalued using a spread corresponding to the new rating category.
3. The bond defaults. In this case a recovery rate is sampled. Often the
recovery rate is assumed to have a beta distribution centered on the
mean recovery rate published by rating agencies (see Table 11.2).
Time Horizon
Regulators require a bank's credit VaR for specific risk to be a ten-day
99% VaR. In practice, this means a bank has two choices. One is to
calculate a one-year 99% VaR in the way we have just outlined and then
use the square-root rule to scale it to a ten-day 99% VaR. If we assume
250 days in a year, this means that the calculated VaR is divided by 5. The
other is to convert the transition matrix in Table 12.1 from a one-year
transition matrix to a ten-day transition matrix and use the procedure we
have just described. to calculate a ten-day loss distribution directly.
Assuming that ratings transitions in successive periods are independent,
the second approach involves finding a matrix B such that
B25 = A
where A is the matrix in Table 12.1. A procedure for doing this is outlined
in Appendix E at the end of the book.
12.7 INTERPRETING CREDIT CORRELATIONS
Care should be taken in interpreting credit correlations. Different ways of
calculating a credit correlation can give quite different answers. We
illustrate this by considering the binomial correlation measure that is
sometimes used by rating agencies and comparing it with the Gaussian
copula correlation measure that underlies Vasicek's model.
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Chapter 12
For two companies, A and B, the binomial correlation measure is the
coefficient of correlation between:
1. A variable that equals 1 if Company A defaults between times 0 and
T and zero otherwise; and
2. A variable that equals 1 if Company B defaults between times 0 and
T and zero otherwise.
The measure is
where P A B ( T ) is the joint probability of A and B defaulting between time
0 and time T, QA{T) is the cumulative probability that Company A will
default by time T, and QB(T) is the cumulative probability that Company
B will default by time T. Typically
depends on T, the length of the
time period considered. Usually it increases as T increases.
In the Gaussian copula model, PAB(T) = M[xA(T), xB(T);
], where
xA(T) =
and xB(T) =
are the transformed times
to default for companies A and B, and
is the Gaussian copula
correlation for the times to default for A and B. The quantity
is the probability that, in a bivariate normal distribution where
the correlation between the variables is
the first variable is less than a
and the second variable is less than b.9 It follows that
This shows that, if QA(T) and QB(T) are known,
can be calculated
from
and vice versa. Usually
is markedly greater than
as
is illustrated by the following example.
Example 12.3
Suppose that the probability of Company A defaulting in one-year period
is 1% and the probability of Company B defaulting in a one-year period
is also 1%. In this case,
=-2.326. If
is
0.20, M(xA(l),xB(l),
) = 0.000337, and equation (12.2) shows that
= 0.024 when T = 1 .
9
An Excel function for calculating
is on the author's website.
Credit Risk Losses and Credit VaR
295
SUMMARY
The credit risk on a loan depends on the probability of default and the
recovery rate (or equivalently the loss given default). The credit risk in a
derivatives transaction is more complicated than that in a loan because
the exposure at the time of default is uncertain. Some derivatives transactions (e.g., written options) are always liabilities and give rise to no
credit risk. Some (e.g., long positions in options) are always assets and
entail significant credit risks. The most difficult types of derivatives
transactions to deal with from a credit risk perspective are those that
may become either assets or liabilities during their life. Examples are
forward contracts and swaps.
The over-the-counter market has developed a number of ways of
mitigating credit risk. The most important of these is netting. This is a
clause in most contracts written by a financial institution stating that, if a
counterparty defaults on one contract it has with the financial institution,
then it must default on all contracts it has with that financial institution.
Another credit mitigation technique is Collateralization. This requires a
counterparty to post collateral. If the value of the contract moves against
the counterparty, more collateral is required. In the event that collateral is
not posted in a timely fashion the contract is closed out using a pre-agreed
procedure for valuation. A third credit mitigation technique is downgrade
trigger. This gives a company the option to close out a contract if the credit
rating of the counterparty falls below a certain level.
Credit VaR can be defined similarly to the way VaR is defined for market
risk. It is the credit loss that will not be exceeded over some time horizon
with a specified confidence level. Basel II calculates credit VaR for the
banking book using a one-factor Gaussian copula model of time to default
that was originally developed by Vasicek. An approach for calculating
credit VaR that is similar to procedures used in the insurance industry is
Credit Risk Plus which was proposed by Credit Suisse Financial Products
in 1997. For specific risk in the trading book most large banks use
CreditMetrics which was proposed by J.P. Morgan in 1997. This involves
simulating rating changes for companies. The correlation between different
companies is handled using a Gaussian copula model for rating changes.
FURTHER READING
Credit Suisse Financial Products, "Credit Risk Management Framework,"
October, 1997.
296
Chapter 12
Finger, C. C, "The One-Factor CreditMetrics Model in the New Basel Capital
Accord," RiskMetrics Journal, Summer 2001.
J.P. Morgan, "CreditMetrics-Technical Document," April, 1997.
QUESTIONS A N D PROBLEMS (Answers at End of Book)
12.1. A bank already has one transaction with a counterparty on its books.
Explain why a new transaction by a bank with a counterparty can have
the effect of increasing or reducing the bank's credit exposure to the
counterparty.
12.2. Suppose that the measure
in equation (12.2) is the same in the real
world and the risk-neutral world. Is the same true of the Gaussian copula
measure
12.3. What is meant by a "haircut" in a Collateralization agreement. A company offers to post its own equity as collateral. How would you respond?
12.4. Explain the difference between Vasicek's model, the Credit Risk Plus
model, and CreditMetrics as far as the following are concerned: (a) when
a credit loss is recognized, and (b) the way in which default correlation is
modeled.
12.5. Suppose that the probability of Company A defaulting during a 2-year
period is 0.2 and the probability of Company B defaulting during this
period is 0.15. If the Gaussian copula measure of default correlation is
0.3, what is the binomial correlation measure?
12.6. Suppose that a financial institution has entered into a swap dependent on
the sterling interest rate with counterparty X and an exactly offsetting
swap with counterparty Y. Which of the following statements are true
and which are false? (a) The total present value of the cost of defaults is
the sum of the present value of the cost of defaults on the contract with X
plus the present value of the cost of defaults on the contract with Y.
(b) The expected exposure in 1 year on both contracts is the sum of the
expected exposure on the contract with X and the expected exposure on
the contract with Y. (c) The 95% upper confidence limit for the exposure
in 1 year on both contracts is the sum of the 95% upper confidence limit
for the exposure in 1 year on the contract with X and the 95% upper
confidence limit for the exposure in 1 year on the contract with Y
Explain your answers.
12.7. A company enters into a 1-year forward contract to sell $100 for AUD 150
The contract is initially at the money. In other words, the forward
exchange rate is 1.50. The 1-year dollar risk-free rate of interest is 5%
per annum. The 1-year dollar rate of interest at which the counterparty can
Credit Risk Losses and Credit VaR
297
borrow is 6% per annum. The exchange rate volatility is 12% per annum.
Estimate the present value of the cost of defaults on the contract. Assume
that defaults are recognized only at the end of the life of the contract.
12.8. Suppose that in Problem 12.7 the 6-month forward rate is also 1.50 and
the 6-month dollar risk-free interest rate is 5% per annum. Suppose
further that the 6-month dollar rate of interest at which the counterparty
can borrow is 5.5% per annum. Estimate the present value of the cost of
defaults assuming that defaults can occur either at the 6-month point or
at the 1-year point? (If a default occurs at the 1-month point, the
company's potential loss is the market value of the contract.)
12.9. "A long forward contract subject to credit risk is a combination of a
short position in a no-default put and a long position in a call subject to
credit risk." Explain this statement.
12.10. Explain why the credit exposure on a pair of offsetting forward contracts
with different counterparties resembles a straddle.
12.11. "When a bank is negotiating a pair of offsetting currency swaps, it should
try to ensure that it is receiving the lower interest rate currency from a
company with a low credit risk." Explain.
ASSIGNMENT QUESTIONS
12.12. Explain carefully the distinction between real-world and risk-neutral
default probabilities. Which is higher? A bank enters into a credit
derivative where it agrees to pay $100 at the end of 1 year if a certain
company's credit rating falls from A to Baa or lower during the year. The
1-year risk-free rate is 5%. Using Table 12.1, estimate a value for the
derivative. What assumptions are you making? Do they tend to overstate
or understate the value of the derivative.
12.13. Suppose that a bank has a total of $10 million of exposures of a certain
type. The one-year probability of default averages 1 % and the recovery
rate averages 40%. The copula correlation parameter is 0.2. Estimate the
1-year 99.5% credit VaR.
12.14. Consider an option on a non-dividend-paying stock where the stock price
is $52, the strike price $50, the risk-free rate is 5%, the volatility is 30%,
and the time to maturity is 1 year. (a) What is the value of the option
assuming no possibility of a default? (b) What is the value of the option
to the buyer if there is a 2% chance that the option seller will default at
maturity? (c) Suppose that, instead of paying the option price up front,
the option buyer agrees to pay the forward value of the option price at
the end of the life of the contract. By how much does this reduce the cost
of defaults to the option buyer in the case where there is a 2% chance of
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Chapter 12
the option seller defaulting? (d) If in case (c) the option buyer has a 1 %
chance of defaulting at the end of the life of the contract, what is the
default risk to the option seller? Discuss the two-sided nature of default
risk in this case and the value of the option to each side.
12.15. Can the existence of downgrade triggers increase default risk? Explain
your answer.
Credit Derivatives
The credit derivatives market has seen huge growth in recent years.
In 2000 the total notional principal for outstanding credit derivative
contracts was about $800 billion. By 2005 this had risen to $12 trillion.
Credit derivatives are contracts where the payoff depends on the
creditworthiness of one or more companies or countries. They allow
companies to trade credit risks in much the same way that they trade
market risks. Banks and other financial institutions, once they had
assumed a credit risk, used to be in the position where they could do
little except wait (and hope for the best). Now they can actively manage
their portfolios of credit risks, keeping some and entering into credit
derivative contracts to protect themselves from others.
13.1 CREDIT DEFAULT SWAPS
The most popular credit derivative is a credit default swap (CDS). This is a
contract that provides insurance against the risk of a default by particular
company. The company is known as the reference entity and a default by
the company is known as a credit event. The buyer of the insurance
obtains the right to sell bonds issued by the company for their face value
when a credit event occurs and the seller of the insurance agrees to buy
the bonds for their face value when a credit event occurs.1 The total face
1
The face value (or par value) of a bond is the principal amount that the issuer will repay
at maturity if it does not default.
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Chapter 13
value of the bonds that can be sold is known as the credit default swap's
notional principal.
The buyer of the CDS makes periodic payments to the seller until the
end of the life of the CDS or until a credit event occurs. These payments
are typically made in arrears every quarter, every half year, or every year.
The settlement in the event of a default involves either physical delivery of
the bonds or a cash payment.
An example will help to illustrate how a typical deal is structured.
Suppose that two parties enter into a five-year credit default swap on
March 1, 2006. Assume that the notional principal is $100 million and the
buyer agrees to pay 90 basis points annually for protection against default
by the reference entity.
The CDS is shown in Figure 13.1. If the reference entity does not
default (i.e., there is no credit event), the buyer receives no payoff and
pays $900,000 on March 1 of each of the years 2007, 2008, 2009, 2010,
and 2011. If there is a credit event a substantial payoff is likely. Suppose
that the buyer notifies the seller of a credit event on June 1, 2009
(a quarter of the way into the fourth year). If the contract specifies
physical settlement, the buyer has the right to sell bonds issued by the
reference entity with a face value of $100 million for $100 million. If the
contract requires cash settlement, an independent calculation agent will
conduct a poll of dealers at a predesignated number of days after the
credit event to determine the mid-market value of the cheapest deliverable bond. Suppose this bond is worth $35 per $100 of face value. The
cash payoff would be $65 million.
The regular quarterly, semiannual, or annual payments from the buyer
of protection to the seller of protection cease when there is a credit event.
However, because these payments are made in arrears, a final accrual
payment by the buyer is usually required. In our example, the buyer would
be required to pay to the seller the amount of the annual payment accrued
between March 1,2009, and June 1,2009 (approximately $225,000), but no
further payments would be required.
The total amount paid per year, as a percent of the notional principal.
to buy protection is known as the CDS spread. Several large banks are
Figure 13.1
Credit default swap.
Credit Derivatives
301
—
Business Snapshot 13.1 Who Bears the Credit Risk?
Traditionally banks have been in the business of making loans and then
bearing the credit risk that the borrower will default. Since 1988 banks have
been reluctant to keep loans to companies with good credit ratings on their
balance sheets. This is because the capital required under Basel I is such that
the expected return from the loans is less attractive than that from investments
in other assets. During the 1990s banks created asset-backed securities to pass
loans (and their credit risk) on to investors. In the late 1990s and early 2000s,
banks have made extensive use of credit derivatives to shift the credit risk in
their loans to other parts of the financial system. (Under Basel II the regulatory
capital for loans to highly rated companies will decline and this may lead to
banks being more willing to keep quality loans on their balance sheet.)
If banks have been net buyers of credit protection, who have been net
sellers? The answer is insurance companies. Insurance companies have not
been regulated in the same way as banks and as a result are sometimes more
willing to bear credit risks than banks.
The result of all this is that the financial institution bearing the credit risk of
a loan is often different from the financial institution that did the original
credit checks. Whether this proves to be good for the overall health of the
financial system remains to be seen.
market makers in the credit default swap market. When quoting on a new
five-year credit default swap on Ford Motor Credit, a market maker
might bid 250 basis points and offer 260 basis points. This means that
the market maker is prepared to buy protection on Ford by paying
250 basis points per year (i.e., 2.5% of the principal per year) and to sell
protection on Ford for 260 basis points per year (i.e., 2.6% of the
principal per year).
As indicated in Business Snapshot 13.1, banks have been the largest
buyers of CDS credit protection and insurance companies have been the
largest sellers.
Credit Default Swaps and Bond Yields
A CDS can be used to hedge a position in a corporate bond. Suppose that
an investor buys a five-year corporate bond yielding 7% per year for its
face value and at the same time enters into a five-year CDS to buy
Protection against the issuer of the bond defaulting. Suppose that the
CDS spread is 2% per annum. The effect of the CDS is to convert the
corporate bond to a risk-free bond (at least approximately). If the bond
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Chapter 13
issuer does not default the investor earns 5% per year (when the CDS
spread is netted against the corporate bond yield). If the bond does
default, the investor earns 5% up to the time of the default. Under the
terms of the CDS, the investor is then able to exchange the bond for its
face value. This face value can be invested at the risk-free rate for the
remainder of the five years.
The n-year CDS spread should be approximately equal to the excess
of the par yield on an n-year corporate bond over the par yield on an
n-year risk-free bond. 2 If it is markedly less than this, an investor can
earn more than the risk-free rate by buying the corporate bond and
buying protection. If it is markedly greater than this, an investor can
borrow at less than the risk-free rate by shorting the corporate bond
and selling CDS protection. These are not perfect arbitrages, but they
are sufficiently good that the CDS spread cannot depart very much
from the excess of the corporate bond par yield over the risk-free par
yield. As we discussed in Section 11.4, a good estimate of the risk-free
rate is the LIBOR/swap rate minus 10 basis points.
The Cheapest-to-Deliver Bond
As explained in Section 11.3, the recovery rate on a bond is defined as the
value of the bond immediately after default as a percent of face value.
This means that the payoff from a CDS is L(l — R), where L is the
notional principal and R is the recovery rate.
Usually a CDS specifies that a number of different bonds can be
delivered in the event of a default. The bonds typically have the same
seniority, but they may not sell for the same percentage of face value
immediately after a default.3 This gives the holder of a CDS a Cheapestto-deliver bond option. When a default happens, the buyer of protection
(or the calculation agent in the event of cash settlement) will review
alternative deliverable bonds and choose for delivery the one that can be
purchased most cheaply. In the context of CDS valuation, R should
therefore be the lowest recovery rate applicable to a deliverable bond.
2
The par yield on an n-year bond is the coupon rate per year that causes the bond to sell
for its par value (i.e., its face value).
3
There are a number of reasons for this. The claim that is made in the event of a default
is typically equal to the bond's face value plus accrued interest. Bonds with high accrued
interest at the time of default therefore tend to have higher prices immediately after
default. The market may also judge that in the event of a reorganization of the company
some bondholders will fare better than others.
Credit Derivatives
303
13.2 CREDIT INDICES
participants in credit derivatives markets have developed indices to track
credit default swap spreads. In 2004 there were agreements between
different producers of indices. This led to some consolidation. Among
the indices now used are:
1. The five- and
spread for 125
2. The five- and
spread for 125
ten-year CDX NA IG indices tracking the credit
investment grade North American companies
ten-year iTraxx Europe indices tracking the credit
investment grade European companies
In addition to monitoring credit spreads, indices provide a way market
participants can easily buy or sell a portfolio of credit default swaps. For
example, an investment bank, acting as market maker might quote the
CDX NA IG five-year index as bid 65 basis points and offer 66 basis
points. An investor could then buy $800,000 of five-year CDS protection
on each of the 125 underlying companies for a total of $660,000 per year.
The investor can sell $800,000 of five-year CDS protection on each of the
125 underlying names for a total of $650,000 per year. When a company
defaults the annual payment is reduced by $660,000/125 = $5,280.4
13.3 VALUATION OF CREDIT DEFAULT SWAPS
Mid-market CDS spreads on individual reference entities (i.e., the average
of the bid and offer CDS spreads quoted by brokers) can be calculated
from default probability estimates. We will illustrate how this is done with
a simple example.
Suppose that the probability of a reference entity defaulting during a
Year conditional on no earlier default is 2%. 5 Table 13.1 shows survival
probabilities and unconditional default probabilities (i.e., default probabilities as seen at time zero) for each of the five years. The probability of a
default during the first year is 0.02 and the probability the reference entity
4
The index is slightly lower than the average of the credit default swap spreads for the
companies in the portfolio. To understand the reason for this, consider two companies,
one with a spread of 1,000 basis points and the other with a spread of 10 basis points. To
buy protection on both companies would cost slightly less than 505 basis points per
company. This is because the 1,000 basis points is not expected to be paid for as long as
the 10 basis points and should therefore carry less weight.
.5 As mentioned in Section 11.2, conditional default probabilities are known as default
intensities or hazard rates.
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Chapter 13
Table 13.1
Unconditional default probabilities
and survival probabilities.
Time
(years)
Default
probability
Survival
probability
1
2
3
4
5
0.0200
0.0196
0.0192
0.0188
0.0184
0.9800
0.9604
0.9412
0.9224
0.9039
will survive until the end of the first year is 0.98. The probability of a
default during the second year is 0.02 x 0.98 = 0.0196 and the probability
of survival until the end of the second year is 0.98 x 0.98 = 0.9604. The
probability of default during the third year is 0.02 x 0.9604 = 0.0192, and
so on.
We will assume that defaults always happen halfway through a year and
that payments on the credit default swap are made once a year, at the end
of each year. We also assume that the risk-free (LIBOR) interest rate is 5%
per annum with continuous compounding and the recovery rate is 40%.
There are three parts to the calculation. These are shown in Tables 13.2,
13.3, and 13.4.
Table 13.2 shows the calculation of the expected present value of the
payments made on the CDS assuming that payments are made at the rate
of s per year and the notional principal is $1. For example, there is a 0.9412
probability that the third payment of s is made. The expected payment is
therefore 0.9412s and its present value is
The
total present value of the expected payments is 4.07045.
Table 13.2 Calculation of the present value of expected payments.
Payment = 5 per annum.
Time
(years)
Probability
of survival
Expected
payment
Discount
factor
PV of expected
payment
1
2
3
4
5
0.9800
0.9604
0.9412
0.9224
0.9039
0.98005
0.96045
0.94125
0.92245
0.90395
0.9512
0.9048
0.8607
0.8187
0.7788
0.93225
0.86905
0.8101s
0.75525
0.70405
Total
4.07045
Credit Derivatives
Table 13.3
305
Calculation of the present value of expected payoff.
Notional principal = $1.
Time
(years)
Probability
of default
Recovery
rate
Expected
payoff ($)
Discount
factor
PV of expected
payoff ($)
0.5
1.5
2.5
3.5
4.5
0.0200
0.0196
0.0192
0.0188
0.0184
0.4
0.4
0.4
0.4
0.4
0.0120
0.0118
0.0115
0.0113
0.0111
0.9753
0.9277
0.8825
0.8395
0.7985
0.0117
0.0109
0.0102
0.0095
0.0088
0.0511
Total
Table 13.3 shows the calculation of the expected present value of the
payoff assuming a notional principal of $1. As mentioned earlier, we are
assuming that defaults always happen halfway through a year. For
example, there is a 0.0192 probability of a payoff halfway through the
third year. Given that the recovery rate is 40% the expected payoff at this
time is 0.0192 x 0.6 x 1 = $0.0115, The present value of the expected
payoff is 0.0115e -0.05x2.5 = $0.0102. The total present value of the expected payoffs is $0.0511.
As a final step we evaluate in Table 13.4 the accrual payment made in
the event of a default. For example, there is a 0.0192 probability that
there will be a final accrual payment halfway through the third year. The
accrual payment is 0.5s. The expected accrual payment at this time is
therefore 0.0192 x 0.5s = 0.0096s. Its present value is 0.0096se -0.05x2.5 =
0.0085s. The total present value of the expected accrual payments
is 0.0426s.
Table 13.4 Calculation of the present value of accrual payment.
Time
(years)
Probability
of default
Expected
accrual payment
Discount
factor
PV of expected
accrual payment
0.5
1.5
2.5
3.5
4.5
0.0200
0.0196
0.0192
0.0188
0.0184
0.01005
0.00985
0.00965
0.00945
0.00925
0.9753
0.9277
0.8825
0.8395
0.7985
0.00975
0.00915
0.00855
0.00795
0.00745
Total
0.04265
306
Chapter 13
From Tables 13.2 and 13.4, we see that the present value of the expected
payments is
4.0704s + 0.0426s = 4.1130s
From Table 13.3 the present value of the expected payoff is $0.0511.
Equating the two, the CDS spread for a new CDS is given by
4.1130s = 0.0511
or s = 0.0124. The mid-market spread should be 0.0124 times the principal
or 124 basis points per year. This example is designed to illustrate the
calculation methodology. In practice, we are likely to find that calculations
are more extensive than those in Tables 13.2 to 13.4 because (a) payments
are often made more frequently than once a year and (b) we might want to
assume that defaults can happen more frequently than once a year.
Marking to Market a CDS
At the time it is negotiated, a CDS like most other swaps is worth close to
zero. At later times it may have a positive or negative value. Suppose, for
example, that the credit default swap in our example had been negotiated
some time ago for a spread of 150 basis points, the present value of the
payments by the buyer would be 4.1130x0.0150 = 0.0617 and the
present value of the payoff would be 0.0511 as above. The value of the
swap to the seller would therefore be 0.0617 - 0.0511, or 0.0106 times the
principal. Similarly, the mark-to market value of the swap to the buyer of
protection would be —0.0106 times the principal.
Estimating Default Probabilities
The default probabilities used to value a CDS should be risk-neutral
default probabilities, not real-world default probabilities (see Section 11.5
for a discussion of the difference between the two). Risk-neutral default
probabilities can be estimated from bond prices or asset swaps, as
explained in Chapter 11. An alternative is to imply them from CDS
quotes. The latter approach is similar to the practice in options markets
of implying volatilities from the prices of actively traded options.
Suppose we change the example in Tables 13.2, 13.3, and 13.4 so that
we do not know the default probabilities. Instead, we know that the midmarket CDS spread for a newly issued five-year CDS is 100 basis points
per year. We can then reverse-engineer our calculations to conclude that
Credit
307
Derivatives
the implied default probability (conditional on no earlier default) is
1.61% per year.6
Binary Credit Default Swaps
A binary credit default swap is structured similarly to a regular credit
default swap except that the payoff is a fixed dollar amount. Suppose, in
the example we have considered in Tables 13.1 to 13.4, that the payoff is
$1 instead of 1 - R dollars and that the swap spread is s. Tables 13.1,
13.2, and 13.4 are the same. Table 13.3 is replaced by Table 13.5. The
CDS spread for a new binary CDS is given by
4.1130s = 0.0852
so that the CDS spread s is 0.0207, or 207 basis points.
How Important is the Recovery Rate?
Whether we use CDS spreads or bond prices to estimate default probabilities, we need an estimate of the recovery rate. However, provided that
we use the same recovery rate for (a) estimating risk-neutral default
probabilities and (b) valuing a CDS, the value of the CDS (or the
estimate of the CDS spread) is not very sensitive to the recovery rate.
This is because the implied probabilities of default are approximately
proportional to 1/(1 - R) and the payoffs from a CDS are proportional
to 1 - R.
Table 13.5
Time
(years)
Probability
of default
Expected
payoff'($)
Discount
factor
PV of expected
payoff ($)
0.5
1.5
2.5
3.5
4.5
0.0200
0.0196
0.0192
0.0188
0.0184
0.0200
0.0196
0.0192
0.0188
0.0184
0.9753
0.9277
0.8825
0.8395
0.7985.
0.0195
0.0182
0.0170
0.0158
0.0147
Total
6
Calculation of the present value of expected payoff from a
binary credit default swap. Principal = $1.
0.0852
Ideally, we would like to estimate a different default probability for each year instead of
a single default intensity. We could do this if we had spreads for 1-, 2-, 3-, 4-, and 5-year
credit default swaps or bond prices.
308
Chapter 13
This argument does not apply to the valuation of a binary CDS. The
probabilities of default implied from a regular CDS are still proportional
to 1/(1 — R). However, for a binary CDS, the payoffs from the CDS are
independent of R. If we have CDS spreads for both a plain vanilla CDS
and a binary CDS, we can estimate both the recovery rate and the
default probability (see Problem 13.23).
The Future of the CDS Market
The market for credit default swaps has grown rapidly in the late 1990s
and early 2000s. Credit default swaps have become important tools for
managing credit risk. A financial institution can reduce its credit exposure
to particular companies by buying protection. It can also use CDSs to
diversify credit risk. For example, if a financial institution has too much
credit exposure to a particular business sector, it can buy protection
against defaults by companies in the sector and at the same time sell
protection against default by companies in other unrelated sectors.
Some market participants believe that the growth of the CDS market
will continue and that it will be as big as the interest rate swap market by
2010. Others are less optimistic. As pointed out in Business Snapshot 13.2,
there is a potential asymmetric information problem in the CDS market
that is not present in other over-the-counter derivatives markets.
13.4 CDS FORWARDS AND OPTIONS
Once the CDS market was well established, it was natural for derivatives
dealers to trade forwards and options on credit default swap spreads.
A forward credit default swap is the obligation to buy or sell a
particular credit default swap on a particular reference entity at a
particular future time T. If the reference entity defaults before time T
the forward contract ceases to exist. Thus, a bank could enter into a
forward contract to sell five-year protection on Ford Motor Credit for
280 basis points starting one year from now. If Ford defaults during the
next year, the bank's obligation under the forward contract ceases to exist.
A credit default swap option is an option to buy or sell a particular
credit default swap on a particular reference entity at a particular future
time T. For example, an investor could negotiate the right to buy five-year
7
The valuation of these instruments is discussed in J.C. Hull and A. White, "The
Valuation of Credit Default Swap Options," Journal of Derivatives, 10, No. 5 (Spring
2003) 40-50.
Credit Derivatives
Business Snapshot 13.2
309
Is the CDS Market a Fair Game?
There is one important difference between credit default swaps and the other
over-the-counter derivatives that we have considered in this book. The other
over-the-counter derivatives depend on interest rates, exchange rates, equity
indices, commodity prices, and so on. There is no reason to assume that any
one market participant has better information than other market participants
about these variables.
Credit default swaps spreads depend on the probability that a particular
company will default during a particular period of time. Arguably some
market participants have more information to estimate this probability than
others. A financial institution that works closely with a particular company by
providing advice, making loans, and handling new issues of securities is likely
:to have more information about the creditworthiness of the company than
another financial institution that has no dealings with the company. Economists refer to this as an asymmetric information problem.
Whether asymmetric information will curtail the expansion of the credit
default swap market remains to be seen. Financial institutions emphasize that
the decision to buy protection against the risk of default by a company is
normally made by a risk manager and is not based on any special information
that many exist elsewhere in the financial institution about the company.
protection on Ford Motor Credit starting in one year for 280 basis points.
This is a call option. If the five-year CDS spread for Ford in one year
turns out to be more than 280 basis points the option will be exercised;
otherwise it will not be exercised. The cost of the option would be paid up
front. Similarly, an investor might negotiate the right to sell five-year
protection on Ford Motor Credit for 280 basis points starting in one year.
This is a put option. If the five-year CDS spread for Ford in one year
turns out to be less than 280 basis points the option will be exercised;
otherwise it will not be exercised. Again the cost of the option would be
Paid up front. Like CDS forwards, CDS options are usually structured so
that they will cease to exist if the reference entity defaults before option
maturity.
An option contract that is sometimes traded in the credit derivatives
market is a call option on a basket of reference entities. If there are m
reference entities in the basket that have not defaulted by the option
maturity, the option gives the holder the right to buy a portfolio of CDSs
on the names for mK basis points, where K is the strike price. In addition,
the holder gets the usual CDS payoff on any reference entities that do
default during the life of the contract.
310
Chapter 13
13.5 TOTAL RETURN SWAPS
A total return swap is a type of credit derivative. It is an agreement to
exchange the total return on a bond (or any portfolio of assets) for
LIBOR plus a spread. The total return includes coupons, interest, and
the gain or loss on the asset over the life of the swap.
An example of a total return swap is a five-year agreement with a
notional principal of $100 million to exchange the total return on a
corporate bond for LIBOR plus 25 basis points. This is illustrated in
Figure 13.2. On coupon payment dates the payer pays the coupons earned
on an investment of $100 million in the bond. The receiver pays interest at a
rate of LIBOR plus 25 basis points on a principal of $100 million. (LIBOR
is set on one coupon date and paid on the next as in a plain vanilla interest
rate swap.) At the end of the life of the swap there is a payment reflecting
the change in value of the bond. For example, if the bond increases in value
by 10% over the life of the swap, the payer is required to pay $10 million
(= 10% of $100 million) at the end of the five years. Similarly, if the bond
decreases in value by 15%, the receiver is required to pay $15 million at the
end of the five years. If there is a default on the bond, the swap is usually
terminated and the receiver makes a final payment equal to the excess of
$100 million over the market value of the bond.
If we add the notional principal to both sides at the end of the life of
the swap, we can characterize the total return swap as follows. The payer
pays the cash flows on an investment of $100 million in the corporate
bond. The receiver pays the cash flows on a $100 million bond paying
LIBOR plus 25 basis points. If the payer owns the corporate bond, the
total return swap allows it to pass the credit risk on the bond to the
receiver. If it does not own the bond, the total return swap allows it to
take a short position in the bond.
Total return swaps are often used as a financing tool. One scenario that
could lead to the swap in Figure 13.2 is as follows. The receiver wants
financing to invest $100 million in the reference bond. It approaches the
payer (which is likely to be a financial institution) and agrees to the swap.
The payer then invests $100 million in the bond. This leaves the receiver in
the same position as it would have been if it had borrowed money at
Figure 13.2
Total return swap.
Credit Derivatives
311
LIBOR plus 25 basis points to buy the bond. The payer retains ownership
of the bond for the life of the swap and faces less credit risk than it would
have done if it had lent money to the receiver to finance the purchase of
the bond, with the bond being used as collateral for the loan. If the
receiver defaults, the payer does not have the legal problem of trying to
realize on the collateral.8
The spread over LIBOR received by the payer is compensation for
bearing the risk that the receiver will default. The payer will lose money
if the receiver defaults at a time when the reference bond's price has
declined. The spread therefore depends on the credit quality of the receiver
and of the bond issuer, and on the default correlation between the two.
There are a number of variations on the standard deal we have
described. Sometimes, instead of a cash payment for the change in the
value of the bond, there is physical settlement where the payer exchanges
the underlying asset for the notional principal at the end of the life of the
swap. Sometimes the change-in-value payments are made periodically
rather than all at the end of the life of the swap.
13.6 BASKET CREDIT DEFAULT SWAPS
In what is referred to as a basket credit default swap there are a number of
reference entities. An add-up basket CDS provides a payoff when any of
the reference entities default. A first-to-default CDS provides a payoff
only when the first default occurs. A second-to-default CDS provides a
payoff only when the second default occurs. More generally, an nth-todefault CDS provides a payoff only when the nth default occurs. Payoffs
are calculated in the same way as for a regular CDS. After the relevant
default has occurred, there is a settlement. The swap then terminates and
there are no further payments by either party.
13.7 COLLATERALIZED DEBT OBLIGATIONS
A collateralized debt obligation (CDO) is a way of creating securities with
widely different risk characteristics from a portfolio of debt instruments.
8
Repos are structured to minimize credit risk in a similar way. A company requiring
short-term funds sells securities to the lender and agrees to buy them back at a later time
at a slightly higher price. The difference between the prices is the interest the lender earns.
If the borrower defaults, the lender keeps the securities. If the lender defaults, the
borrower keeps the funds.
312
Chapter 13
Figure 13.3
Collateralized debt obligation.
An example is shown in Figure 13.3, where four types of securities (or
tranches) are created from a portfolio of bonds. The first tranche has 5% of
the total bond principal and absorbs all credit losses from the portfolio
during the life of the CDO until they have reached 5% of the total bond
principal. The second tranche has 10% of the principal and absorbs all
losses during the life of the CDO in excess of 5% of the principal up to a
maximum of 15% of the principal. The third tranche has 10% of the
principal and absorbs all losses in excess of 15% of the principal up to a
maximum of 25% of the principal. The fourth tranche has 75% of the
principal and absorbs all losses in excess of 25% of the principal. The
yields in Figure 13.3 are the rates of interest paid to tranche holders. These
rates are paid on the balance of the principal remaining in the tranche after
losses have been paid. Consider the first tranche. Initially the return of
35% is paid on the whole amount invested by the tranche holders. But after
losses equal to 1 % of the total bond principal have been experienced, the
tranche holders have lost 20% of their investment and the return is paid on
only 80% of the original amount invested.9 Tranche 1 is referred to as the
9
When a bond with principal Q defaults and a recovery of QR is made, the usual
arrangement is that a loss of (1 - Q)R is sustained by the most junior tranche. An
amount QR is paid to the most senior tranche and this tranche's principal is reduce
by QR.
Credit Derivatives
313
equity tranche. A default loss of 2.5% on the bond portfolio translates into
a loss of 50% of the tranche's principal. Tranche 4 is usually given an Aaa
rating. Defaults on the bond portfolio must exceed 25% before the holders
of this tranche are responsible for any credit losses.
The creator of the CDO normally retains the equity tranche and sells
the remaining tranches in the market. A CDO provides a way of creating
high-quality debt from average-quality (or even low-quality) debt.
Synthetic CDOs
The CDO in Figure 13.3 is referred to as a cash CDO. An alternative
structure which has become popular is a synthetic CDO, where the creator
of the CDO sells a portfolio of credit default swaps to third parties. It
then passes the default risk on to the synthetic CDO's tranche holders.
Analogously to Figure 13.3, the first tranche might be responsible for the
payoffs on the credit default swaps until they have reached 5% of the total
notional principal; the second tranche might be responsible for the payoffs between 5% and 15% of the total notional principal; and so on. The
income from the credit default swaps is distributed to the tranches in a
way that reflects the risk they are bearing. For example, the first tranche
might get 3,000 basis points; the second tranche 1,000 basis points, and
so on. As with a cash CDO, losses on defaults would be netted against
the principal to determine the amount on which interest is paid.
Single-Tranche Trading
In Section 13.2 we discussed the portfolios of 125 companies that are used
to generate CDX and iTraxx indices. The market uses these portfolios to
define standard CDO tranches. The trading of these standard tranches is
known as single-tranche trading. A single-tranche trade is an agreement
where one side agrees to sell protection against losses on a tranche and the
other side agrees to buy the protection. The tranche is not part of a
synthetic CDO, but cash flows are calculated in the same way as if it were
part of a synthetic CDO. The tranche is referred to as "unfunded" because
it has not been created by selling credit default swaps or buying bonds.
In the case of the CDX NA IG index, the equity tranche covers losses
between 0% and 3% of the principal. The second tranche, which is
referred to as the mezzanine tranche, covers losses between 3% and 7%.
The remaining tranches cover losses from 7% to 10%, 10% to 15%, and
15% to 30%. In the case of the iTraxx Europe index, the equity tranche
covers losses between 0% and 3%. The mezzanine tranche covers losses
314
Chapter 13
Table 13.6 Five-year CDX IG NA and iTraxx Europe tranches on
August 30, 2005. Quotes are in basis points except for 0-3% tranche.
Source: Reuters
CDX IG NA
Tranche
Quote
iTraxx Europe
Tranche
Quote
0-3%
40%
3-7%
127
7-10%
35.5
10-15%
20.5
15-30%
9.5
0-3%
24%
3-6%
81
6-9%
26.5
9-12%
15
12-22%
9
between 3% and 6%. The remaining tranches cover losses from 6% to 9%,
9% to 12%, and 12% to 22%.
Table 13.6 shows the mid-market quotes for the five-year CDX and
iTraxx tranches on August 30, 2005. On that date the CDX index level was
50 basis points and the iTraxx index was 36.375 basis points. For example,
the mid-market price of mezzanine protection for the CDX IG NA was
127 basis points per year, while that for iTraxx Europe was 81 basis points
per year. Note that the equity tranche is quoted differently from the other
tranches. The market quote of 40% for CDX means that the protection
seller receives an initial payment of 40% of the principal plus a spread of
500 basis points per year. Similarly, the market quote of 24% for iTraxx
means that the protection seller receives an initial payment of 24% of the
principal plus a spread of 500 basis points per year.
13.8 VALUATION OF A BASKET CDS AND CDO
The spread for an nth-to-default CDS or the tranche of a CDO is critically
dependent on default correlation. Suppose that a basket of 100 reference
entities is used to define a five-year nth-to-default CDS and that each
reference entity has a risk-neutral probability of defaulting during the five
years equal to 2%. When the default correlation between the reference
entities is zero, the binomial distribution shows that the probability of one
or more defaults during the five years is 86.74% and the probability often
or more defaults is 0.0034%. A first-to-default CDS is therefore quite
valuable, whereas a tenth-to-default CDS is worth almost nothing.
As the default correlation increases the probability of one or more
defaults declines and the probability of ten or more defaults increases.
In the limit where the default correlation between the reference entities
Credit Derivatives
Business Snapshot 13.3
315
Correlation Smiles
Credit derivatives dealers imply default correlations from the spreads on
tranches. The compound correlation is the correlation that prices a particular
tranche correctly. The base correlation is the correlation that prices all tranches
up to a certain level of seniority correctly. If all implied correlations were the
same, we could deduce that market prices are consistent with the one-factor
Gaussian copula model for time to default. In practice, we find that compound
correlations exhibit a "smile" with the correlations for the most junior (equity)
and senior tranches higher than those for intermediate tranches. The base
correlations exhibit a "skew" where the correlation increases with the level of
seniority considered.
is perfect, the probability of one or more defaults equals the probability
of ten or more defaults and is 2%. This is because in this extreme
situation the reference entities are essentially the same. Either they all
default (with probability 2%) or none of them default (with probability 98%).
The valuation of a tranche of a CDO is similarly dependent on default
correlation. If the correlation is low, the junior equity tranche is very risky
and the senior tranches are very safe. As the default correlation increases,
the junior tranches become less risky and the senior tranches become more
risky. In the limit where the default correlation is perfect the tranches are
equally risky.
Using the Gaussian Copula Model of Time to Default
The one-factor Gaussian copula model of time to default presented in
Section 6.5 has become the standard market model for valuing an nth-todefault CDS or a tranche of a CDO.
Consider a portfolio of N companies, each having a probability Q(T)
of defaulting by time T. From equation (6.11), the probability of
default, conditional on the level of the factor F, is
The trick to valuing an nth-to-default CDS or a CDO is to calculate
expected cash flows conditional on F and then integrate over F. The
advantage of this is that, conditional on F, defaults are independent. The
316
Chapter 13
probability of exactly k defaults by time T, conditional on F, is
Derivatives dealers calculate the implied copula correlation
in
equation (13.1) from the spreads quoted in the market for tranches of
CDOs and tend to quote these rather than the spreads themselves (see
Business Snapshot 13.3). This is similar to the practice in options markets
of quoting Black-Scholes implied volatilities rather than dollar prices.
SUMMARY
Financial institutions use credit derivatives to actively manage their
credit risks. They use them to transfer credit risk from one company
to another and to diversify credit risk by swapping one type of exposure
for another.
The most common credit derivative is a credit default swap. This is a
contract where one company buys insurance against another company
defaulting on its obligations. The payoff is usually the difference between
the face value of a bond issued by the second company and its value
immediately after a default. Credit default swaps can be analyzed by
calculating the present value of the expected payments and the present
value of the expected payoff.
A forward credit default swap is an obligation to enter into a particular
credit default swap on a particular date. A credit default swap option is the
right to enter into a particular credit default swap on a particular date.
Both cease to exist if the reference entity defaults before the date.
A total return swap is an instrument where the total return on a
portfolio of credit-sensitive assets is exchanged for LIBOR plus a spread.
Total return swaps are often used as financing vehicles. A company
wanting to purchase a portfolio of bonds approaches a financial institution, who buys the bonds on its behalf. The financial institution then
enters into a total return swap where it pays the return on the bonds to
the company and receives LIBOR plus a spread. The advantage of this
type of arrangement is that the financial institution reduces its exposure
to defaults by the company.
An nth-to-default CDS is defined as a CDS that pays off when the nth
default occurs in a portfolio of companies. In a collateralized debt
obligation, a number of different securities are created from a portfolio
Credit Derivatives
317
of corporate bonds or commercial loans. There are rules for determining
how credit losses are allocated to the securities. The result of the rules is
that securities with both very high and very low credit ratings are created
from the portfolio. A synthetic collateralized debt obligation creates a
similar set of securities from credit default swaps. The standard market
model for pricing both an nth-to-default CDS and tranches of a CDO is
the one-factor Gaussian copula model for time to default.
FURTHER READING
Andersen, L., J. Sidenius, and S. Basu, "All Your Hedges in One Basket," Risk,
November 2003.
Andersen, L., and J. Sidenius, "Extensions to the Gaussian Copula: Random
Recovery and Random Factor Loadings," Journal of Credit Risk, 1, No. 1
(Winter 2004): 29-70.
Das, S., Credit Derivatives: Trading & Management of Credit & Default Risk.
Singapore: Wiley, 1998.
Hull, J. C, and A. White, "Valuation of a CDO and nth to Default Swap
without Monte Carlo Simulation," Journal of Derivatives, 12, No. 2 (Winter
2004): 8-23.
Hull, J. C, and A. White, "The Perfect Copula," Working Paper, University of
Toronto.
Laurent, J.-P., and J: Gregory, "Basket Default Swaps, CDOs and Factor
Copulas," Working Paper, ISFA Actuarial School, University of Lyon, 2003.
Li, D. X., "On Default Correlation: A Copula Approach," Journal of Fixed
Income, March 2000: 43-54.
Tavakoli, J. M., Credit Derivatives: A Guide to Instruments and Applications.
New York: Wiley, 1998.
Schonbucher, P. J., Credit Derivatives Pricing Models. Wiley, 2003.
QUESTIONS A N D PROBLEMS (Answers at End of Book)
13.1. Explain the difference between a regular credit default swap and a binary
credit default swap.
13.2. A credit default swap requires a semiannual payment at the rate of 60 basis
points per year. The principal is $300 million and the credit default swap is
settled in cash. A default occurs after 4 years and 2 months, and the
calculation agent estimates that the price of the cheapest deliverable bond
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Chapter 13
is 40% of its face value shortly after the default. List the cash flows and
their timing for the seller of the credit default swap.
13.3. Explain the two ways a credit default swap can be settled.
13.4. Explain how a cash CDO and a synthetic CDO are created.
13.5. Explain what a first-to-default credit default swap is. Does its value
increase or decrease as the default correlation between the companies
in the basket increases? Explain your answer.
13.6. Explain the difference between risk-neutral and real-world default
probabilities.
13.7. Explain why a total return swap can be useful as a financing tool.
13.8. Suppose that the risk-free zero curve is flat at 7% per annum with
continuous compounding and that defaults can occur halfway through
each year in a new 5-year credit default swap. Suppose that the recovery
rate is 30% and the default probabilities each year conditional on no
earlier default is 3%. Estimate the credit default swap spread. Assume
payments are made annually.
13.9. What is the value of the swap in Problem 13.8 (per dollar of notional
principal) to the protection buyer if the credit default swap spread is
150 basis points?
13.10. What is the credit default swap spread in Problem 13.8 if it is a binary
CDS?
13.11. How does a 5-year nth-to-default credit default swap work? Consider a
basket of 100 reference entities where each reference entity has a probability of defaulting in each year of 1%. As the default correlation
between the reference entities increases, what would you expect to
happen to the value of the swap when (a) n = 1 and (b) n = 25. Explain
your answer.
13.12. How is the recovery rate of a bond usually defined?
13.13. Show that the spread for a new plain vanilla CDS should be (1 - R)
times the spread for a similar new binary CDS, where R is the recovery
rate.
13.14. A company enters into a total return swap where it receives the return on
a corporate bond paying a coupon of 5% and pays LIBOR. Explain the
difference between this and a regular swap where 5% is exchanged for
LIBOR.
13.15. Explain how forward contracts and options on credit default swaps are
structured.
13.16. "The position of a buyer of a credit default swap is similar to the position
of someone who is long a risk-free bond and short a corporate bondExplain this statement.
Credit
Derivatives
319
13.17. Why is there a potential asymmetric information problem in credit
default swaps?
13.18. Does valuing a CDS using real-world default probabilities rather than
risk-neutral default probabilities overstate or understate the value of the
protection? Explain your answer.
13.19. What is the difference between a total return swap and an asset swap?
13.20. Suppose that in a one-factor Gaussian copula model the 5-year probability of default for each of 125 names is 3% and the pairwise copula
correlation is 0.2. Calculate, for factor values of —2, — 1, 0, 1, and 2,
(a) the default probability conditional on the factor value and (b) the
probability of more than 10 defaults conditional on the factor value.
13.21. What is a CDO squared? How about a CDO cubed?
ASSIGNMENT QUESTIONS
13.22. Suppose that the risk-free zero curve is flat at 6% per annum with
continuous compounding and that defaults can occur at times 0.25 years,
0.75 years, 1.25 years, and 1.75 years in a 2-year plain vanilla credit
default swap with semiannual payments. Suppose that the recovery rate is
20% and the unconditional probabilities of default (as seen at time zero)
are 1% at times 0.25 years and 0.75 years, and 1.5% at times 1.25 years
and 1.75 years. What is the credit default swap spread? What would the
credit default spread be if the instrument were a binary credit default
swap?
13.23. Assume that the default probability for a company in a year, conditional
on no earlier defaults is and the recovery rate is R. The risk-free interest
rate is 5% per annum. Default always occurs halfway through a year.
The spread for a 5-year plain vanilla CDS where payments are made
annually is 120 basis points and the spread for a 5-year binary CDS
where payments are made annually is 160 basis points. Estimate R and
13.24. Explain how you would expect the yields offered on the various tranches
in a CDO to change when the correlation between the bonds in the
portfolio increases.
13.25. Suppose that (a) the yield on a 5-year risk-free bond is 7%, (b) the yield on
a 5-year corporate bond issued by company X is 9.5%, and (c) a 5-year
credit default swap providing insurance against company X defaulting
costs 150 basis points per year. What arbitrage opportunity is there in this
situation? What arbitrage opportunity would there be if the credit default
spread were 300 basis points instead of 150 basis points? Give two reasons
why arbitrage opportunities such as those you have identified are less than
perfect.
Operational Risk
In 1999, bank supervisors announced plans to assign capital for operational risk in the new Basel II regulations. This met with some opposition
from banks. The chairman and CEO of one major international bank
described it as "the dopiest thing I have ever seen". However, bank supervisors persisted. They argued that operational risk was a major issue for
banks. They pointed out that during a ten-year period more than 100
operational risk losses, each exceeding $100 million, had occurred. Some of
these losses, listed by the categories used by the Bank for International
Settlements, are:
Internal fraud: Allied Irish Bank, Barings, and Daiwa lost $700 million,
$1 billion, and $1.4 billion, respectively, from fraudulent trading.
External fraud: Republic New York Corp. lost $611 million because of
fraud committed by a custodial client.
Employment practices and workplace safety: Merrill Lynch lost $250 million in a legal settlement regarding gender discrimination.
Clients, products, & business practices: Household International lost
$484 million from improper lending practices; Providian Financial Corporation lost $405 million from improper sales and billing practices.
Damage to physical assets: Bank of New York lost $140 million because of
damage to its facilities related to the September 11, 2001, terrorist attack.
Business disruption and system failures: Solomon Brothers lost $303 million from a change in computing technology.
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Execution, delivery and process management: Bank of America and Wells
Fargo Bank lost $225 million and $150 million, respectively, from systems
integration failures and transactions processing failures.
Most banks have always had some framework in place for managing
operational risk. However, the prospect of new capital requirements has
led them to greatly increase the resources they devote to measuring and
monitoring operational risk.
It is much more difficult to quantify operational risk than credit or
market risk. Operational risk is also more difficult to manage. Banks make
a conscious decision to take a certain amount of credit and market risk,
and there are many traded instruments that can be used to reduce these
risks. Operational risk, by contrast, is a necessary part of doing business.
An important part of operational risk management is identifying the types
of risks that are being taken and which should be insured against. There is
always a danger that a huge loss will be incurred from taking an operational risk that ex ante was not even recognized as a risk.
It might be thought that a loss such as that which brought down
Barings Bank was a result of market risk because it was movements in
market variables that led to it. However, it should be classified as
operational risk because it involved fraud by one of its traders, Nick
Leeson (see Business Snapshot 2.4). Suppose there was no fraud. If it was
the bank's policy to let traders take huge risks, then the loss would be
classified as market risk. But if this was not the bank's policy and there
was a breakdown in its controls, then it would be classified as operational
risk. Operational risk losses are often contingent on market movements.
If the market had moved in Leeson's favor, there would have been no
loss. The fraud and breakdown in the bank's control systems would
probably never have come to light.
There are some parallels between the operational risk losses of banks
and the losses of insurance companies. Insurance companies face a small
probability of a large loss arising from a hurricane, earthquake, or other
natural disaster. Similarly, banks face a small probability of a large
operational risk loss. But there is one important difference. When insurance companies lose a large amount of money because of a natural
disaster, all companies in the industry tend to be affected and premiums
rise the next year to cover losses. Operational risk losses tend to affect only
one bank. Since it operates in a competitive environment, the bank does
not have the luxury of increasing prices for the services it offers during the
following year.
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323
14.1 WHAT IS OPERATIONAL RISK?
There are many different ways in which operational risk can be defined. It
is tempting to consider operational risk as a residual risk and define it as
any risk faced by a bank that is not market risk or credit risk. To produce
an estimate of operational risk, we could then look at the bank's financial
statements and remove from the income statement (a) the impact of credit
losses and (b) the profits or losses from market risk exposure. The variation
in the resulting income would then be attributed to operational risk.
Most people agree that this definition of operational risk is too broad.
It includes the risks associated with entering new markets, developing new
products, economic factors, and so on. Another possible definition is that
operational risk, as its name implies, is the risk arising from operations.
This includes the risk of mistakes in processing transactions, making
payments, etc. This definition of risk is too narrow. It does not include
major risks such as the "rogue trader" risk.
We can distinguish between internal risks and external risks. Internal
risks are those over which the company has control. The company chooses
whom it employs, what computer systems it develops, what controls are in
place, and so on. Some people define operational risks as all internal risks.
Operational risk then includes more than just the risk arising from operations. It includes risks arising from inadequate controls such as the rogue
trader risk and the risks of other sorts of employee fraud.
Regulators favor including more than just internal risks in their definition of operational risk. They include the impact of external events, such as
natural disasters (e.g., a fire or an earthquake that affects the bank's
operations), political or regulatory risk (e.g., being prevented from operating in a foreign country by that country's government), security breaches,
and so on. All of this is reflected in the following definition of operational
risk produced by the Basel Committee on Banking Supervision in 2001:
The risk of loss resulting from inadequate or failed internal processes,
people, and systems or from external events.
Note that this definition includes legal risk, but does not include reputation risk or the risk resulting from strategic decisions.
Some operational risks result in increases in the bank's operating cost
or decreases in its revenue. Other operational risks interact with credit
and market risk. For example, when mistakes are made in a loan's
documentation, it is usually the case that losses result if and only if
the counterparty defaults. When a trader exceeds limits and misreports
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positions, losses result if and only if the market moves against the
trader.
14.2 DETERMINATION OF REGULATORY CAPITAL
Banks have three alternatives for determining operational risk regulatory
capital. The simplest approach is the basic indicator approach. Under this
approach, operational risk capital is set equal to 15% of annual gross
income over the previous three years. Gross income is defined as net
interest income plus noninterest income.1 A slightly more complicated
approach is the standardized approach, in which a bank's activities are
divided into eight business lines: corporate finance, trading and sales,
retail banking, commercial banking, payment and settlement, agency
services, asset management, and retail brokerage. The average gross
income over the last three years for each business line is multiplied by a
"beta factor" for that business line and the result summed to determine
the total capital. The beta factors are shown in Table 14.1. The third
alternative is the advanced measurement approach (AMA), in which the
operational risk regulatory capital requirement is calculated by the bank
internally using qualitative and quantitative criteria.
The Basel Committee has listed conditions that a bank must satisfy in
order to use the standardized approach or the AMA approach. It expects
large internationally active banks to move toward adopting the AMA
Table 14.1
Beta factors in standardized
approach.
Business line
Corporate finance
Trading and sales
Retail banking
Commercial banking
Payment and settlement
Agency services
Asset management
Retail brokerage
Beta factor
18%
18%
12%
15%
18%
15%
12% . ..
12%
1
Net interest income is the excess of income earned on loans over interest paid on
deposits and other instruments that are used to fund the loans (see Section 1.3).
Operational
325
Risk
approach through time. To use the standardized approach, a bank must
satisfy the following conditions:
1. The bank must have an operational risk management function that
is responsible for identifying, assessing, monitoring, and controlling
operational risk.
2. The bank must keep track of relevant losses by business line and
must create incentives for the improvement of operational risk.
3. There must be regular reporting of operational risk losses throughout the bank.
4. The bank's operational risk management system must be well
documented.
5. The bank's operational risk management processes and assessment
system must be subject to regular independent reviews by internal
auditors. It must also be subject to regular review by external
auditors or supervisors or both.
To use the AMA approach, the bank must satisfy additional requirements. It must be able to estimate unexpected losses based on an analysis
of relevant internal and external data, and scenario analyses. The bank's
system must be capable of allocating economic capital for operational risk
Figure 14.1
Calculation of VaR for operational risk.
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across business lines in a way that creates incentives for the business lines
to improve operational risk management.
The objective of banks using the AMA approach for operational risk is
analogous to their objectives when they attempt to quantify credit risk.
They would like to produce a probability distribution of losses such as
that shown in Figure 14.1. Assuming that they can convince regulators
that their expected operational risk cost is incorporated into their pricing
of products, capital is assigned to cover unexpected costs. The confidence
level is 99.9% and the time horizon is one year.
14.3 CATEGORIZATION OF OPERATIONAL RISKS
The Basel Committee on Bank Supervision has identified seven categories
of operational risk.2 These are:
1. Internal fraud: Acts of a type intended to defraud, misappropriate
property or circumvent regulations, the law, or company policy
(excluding diversity or discrimination events which involve at least
one internal party). Examples include intentional misreporting of
positions, employee theft, and insider trading on an employee's own
account.
2. External fraud: Acts by third party of a type intended to defraud,
misappropriate property or circumvent the law. Examples include
robbery, forgery, check kiting, and damage from computer hacking.
3. Employment practices and workplace safety: Acts inconsistent with
employment, health or safety laws or agreements, or which result in
payment of personal injury claims, or claims relating to diversity or
discrimination issues. Examples include workers compensation
claims, violation of employee heath and safety rules, organized
labor activities, discrimination claims, and general liability (e.g., a
customer slipping and falling at a branch office).
4. Clients, products, and business practices: Unintentional or negligent
failure to meet a professional obligation to specific clients (including
fiduciary and suitability requirements), or from the nature or design
of a product. Examples include fiduciary breaches, misuse of
confidential customer information, improper trading activities on
2
See Basel Committee on Bank Supervision, "Sound Practices for the Management and
Supervision of Operational Risk," Bank for International Settlements, July 2002.
Operational Risk
327
the bank's account, money laundering, and the sale of unauthorized
products.
5. Damage to physical assets: Loss or damage to physical assets from
natural disasters or other events. Examples include terrorism,
vandalism, earthquakes, fires, and floods.
6. Business disruption and system failures: Disruption of business or
system failures. Examples include hardware and software failures,
telecommunication problems, and utility outages.
7. Execution, delivery, and process management: Failed transaction
processing or process management, and relations with trade counterparties and vendors. Examples include data entry errors, collateral
management failures, incomplete legal documentation, unapproved
access given to clients accounts, nonclient counterparty misperformance, and vendor disputes.
Banks must assess their exposure to each type of risk for each of the
eight business lines listed in Table 14.1. Ideally this will lead to a result
where VaR is estimated for each of 7 x 8 = 56 risk-type/business-line
combinations.
14.4 LOSS SEVERITY AND LOSS FREQUENCY
There are two distributions that are important in estimating potential
operational risk losses. One is the loss frequency distribution and the other
is the loss severity distribution. The loss frequency distribution is the
distribution of the number of losses observed during the time horizon
(usually one year). The loss severity distribution is the distribution of the
size of a loss, given that a loss occurs. It is usually assumed that loss
severity and loss frequency are independent.
For loss frequency, the natural probability distribution to use is a
Poisson distribution. This distribution assumes that losses happen randomly through time so that in any short period of time
there is a
probability
of a loss being sustained. The probability of n losses in
time T is
The parameter can be estimated as the average number of losses per unit
time. For example, if during a 10-year period there were a total 12 losses,
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Chapter 14
then is 1.2 per year or 0.1 per month. A Poisson distribution has the
property that the mean frequency of losses equals the variance of the
frequency of losses.3
For the loss severity probability distribution, a lognormal probability
distribution is often used. The parameters of this probability distribution
are the mean and standard deviation of the logarithm of the loss.
The loss frequency distribution must be combined with the loss
severity distribution for each loss type and business line to determine a
total loss distribution. Monte Carlo simulation can be used for this
purpose.4 As mentioned earlier, the usual assumption is that loss severity
is independent of loss frequency. On each simulation trial, we proceed as
follows:
1. We sample from the frequency distribution to determine the number
of loss events (= n).
2. We sample n times from the loss severity distribution to determine
the loss experienced for each loss event
3. We determine the total loss experienced
When many simulation trials are used, we obtain a total loss distribution.
Figure 14.2 illustrates the procedure. In this example the expected loss
frequency is 3 per year and the loss severity is drawn from a lognormal
distribution. The logarithm of a loss ($ millions) is assumed to have a
mean of 0 and a standard deviation of 0.4. The Excel worksheet used to
produce Figure 14.2 is on the author's website.
Data Issues
Unfortunately there is usually relatively little historical data available
within a bank to estimate loss severity and loss frequency distributions.
Many banks have not kept records of losses arising from different types
of operational risks for different business lines. As a result of regulatory
pressure, they are starting to do so, but it may be some time before a
reasonable amount of historical data is available. It is interesting to
compare operational risk losses with credit risk losses in this respect.
3
If the mean frequency is greater than the variance of the frequency, a binomial
distribution may be more appropriate. If the mean frequency is less than the variance, a
negative binomial distribution (mixed Poisson distribution) may be more appropriate4
Combining the loss severity and loss frequency distribution is a very common problem
in insurance. Apart from Monte Carlo simulation, two approaches that are used are
Panjer's algorithm and fast Fourier transforms. See H.H. Panjer, "Recursive Evaluation
of a Family of Compound Distributions," ASTIN Bulletin, 12 (1981), 22-29.
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Operational Risk
Figure 14.2
Calculation of loss distribution from loss frequency and
loss severity.
Traditionally banks have done a much better job at documenting their
credit risk losses than their operational risk losses. Moreover, in the case
of credit risks, a bank can rely on a wealth of information published by
credit-rating agencies to assess probabilities of default and expected losses
given default. Similar data on operational risk has not in the past been
collected in such a systematic way. It may also be a problem that banks
sometimes conceal a large operational risk loss from the market because
they feel it will damage their reputation.
As indicated above, the Poisson distribution is often used for loss
frequency and the lognormal distribution is often used for loss severity.
Available data is usually used to estimate the parameters of these
distributions. The loss frequency distribution should be estimated from
the bank's own data as far as possible. For the loss severity distribuion, regulators encourage banks to use their own data in conjunction
with external data. There are two sources of external data. The first is
data obtained through sharing arrangements between banks. (The
insurance industry has had mechanisms for sharing loss data for many
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years and banks are now beginning to do this as well.) The second is
publicly available data that has been collected in a systematic way by
data vendors.
Both internal and external historical data must be adjusted for inflation. In addition, a scale adjustment should be made to external data. If a
bank with a revenue of $10 billion reports a loss of $8 million, how
should the loss be scaled for a bank with a revenue of $5 billion? A
natural assumption is that a similar loss for a bank with a revenue of
$5 billion would be $4 million. This estimate is probably too small. For
example, research by Shih et al. suggests that the effect of firm size on the
size of a loss experience is relatively small.5 Their estimate is
Estimated loss for Bank A
= Observed loss for Bank B x
where = 0.23. This means that in our example the bank with a revenue
of $5 billion would experience a loss of 8 x 0.50.23 = 6.82 million.
After the appropriate scale adjustment, data obtained through sharing
arrangements with other banks can be merged with the bank's own data
to obtain a larger sample for determining the loss severity distribution.
Public data purchased from data vendors cannot be used in this way
because it is subject to biases. For example:
1. Only large losses are publicly reported, and the larger the loss, the
more likely it is to be reported.
2. Institutions with weak controls are more likely to be represented in
the database because they suffer more losses. Moreover, their losses
tend to be larger.
Public data is most useful for determining relative loss severity. Suppose
that a bank has good information on the mean and standard deviation of
its loss severity distribution for internal fraud in corporate finance, but not
for external fraud in corporate finance or for internal fraud in trading and
sales. Suppose that the mean and standard deviation of its internal loss
severity distribution for internal fraud in corporate finance are $50,000 and
$30,000. Suppose further that external data indicates that for external
fraud in corporate finance the mean severity is twice that for internal fraud
5
See J. Shih, A. Samad-Khan, and P. Medapa, "Is the Size of an Operational Loss
Related to Firm Size," Operational Risk, January 2000. Whether Shih et a/.'s results apply
to legal risks is debatable. It often seems that the size of a settlement in a large lawsuit
against a bank is governed by how much the bank can afford.
Operational Risk
331
in corporate finance and the standard deviation of the severity is 1.5 times
as great. In the absence of a better alternative, the bank might assume that
jts own severity for external fraud in corporate finance has a mean of
2 x 50,000 = $100,000 and a standard deviation of severity equal to
1.5 x 30,000 = $45,000. Similarly, if the external data indicates that the
mean severity for internal fraud in trading and sales is 2.5 times that for
internal fraud in corporate finance and the standard deviation is twice as
great, the bank might assume that its own severity for internal fraud in
trading and sales has a mean of 2.5 x 50,000 = $100,000 and a standard
deviation of 2 x 30,000 = $60,000.
Scenario Analysis
Since historical data is relatively difficult to obtain, regulators encourage
banks to use scenario analyses in addition to internal and external loss
data. This involves using managerial judgement to generate scenarios
where large losses occur. Managers estimate the loss frequency parameter
associated with each scenario and the parameters of the loss severity
distribution. The advantage of scenario analysis is that it can include
losses that the financial institution has never experienced, but, in the
judgement of senior management, could occur. It reflects the controls
in place in the bank and the type of business it is currently doing.
One advantage of the scenario analysis approach is that it leads to
management thinking actively and creatively about potential adverse
events. This can have a number of benefits. In some cases strategies for
responding to an event so as to minimize its severity are likely to be
developed. In other cases, proposals may be made for reducing the
probability of the event occurring at all.
The main drawback of scenario analysis is that it requires a great deal
of senior management time. It seems likely that standard scenarios will be
developed by consultants and by banks themselves to make the process
less of a burden.
14.5 FORWARD LOOKING APPROACHES
Risk managers should try to be proactive in preventing losses from
occurring. One approach is to monitor what is happening at other banks
and try and learn from their mistakes. When a $700 million rogue trader
loss happened at a Baltimore subsidiary of Allied Irish Bank in 2002, risk
managers throughout the world studied the situation carefully and asked:
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Chapter 14
Business Snapshot 14.1
The Hammersmith and Fulham Story
Between 1987 to 1989 the London Borough of Hammersmith and Fulham in
Great Britain entered into about 600 interest rate swaps and related instruments
with a total notional principal of about 6 billion pounds. The transactions
appear to have been entered into for speculative rather than hedging purposes.
The two employees of Hammersmith and Fulham that were responsible for the
trades had only a sketchy understanding of the risks they were taking and how
the products they were trading worked.
By 1989, because of movements in sterling interest rates, Hammersmith and
Fulham had lost several hundred million pounds on the swaps. To the banks
on the other side of the transactions, the swaps were worth several hundred
million pounds. The banks were concerned about credit risk. They had entered
into offsetting swaps to hedge their interest rate risks. If Hammersmith and
Fulham defaulted, they would still have to honor their obligations on the
offsetting swaps and would take a huge loss.
What actually happened was not a default. Hammersmith and Fulham's
auditor asked to have the transactions declared void because Hammersmith
and Fulham did not have the authority to enter into the transactions. The
British courts agreed. The case was appealed and went all the way to the
House of Lords, Britain's highest court. The final decision was that Hammersmith and Fulham did not have the authority to enter into the swaps, but that
they ought to have the authority to do so in the future for risk management
purposes. Needless to say, banks were furious that their contracts were overturned in this way by the courts.
"Could this happen to us?" Business Snapshot 14.1 describes a situation
concerning a British local authority in the late 1980s. It immediately led
to all banks instituting procedures for checking that counterparties had
the authority to enter into derivatives transactions.
Causal Relationships
Operational risk managers should try and establish causal relations
between decisions taken and operational risk losses. Does increasing the
average educational qualifications of employees reduce losses arising from
mistakes in the way transactions are processed? Will a new computet
system reduce the probabilities of losses from system failures? Are operational risk losses correlated with the employee turnover rate? If so, can
they be reduced by measures taken to improve employee retention? Can
the risk of a rogue trader be reduced by the way responsibilities are divide
between different individuals and by the way traders are motivated?
Operational Risk
333
One approach to establishing causal relationships is statistical. If we
look at 12 different locations where a bank operates and find a high
negative correlation between the education of back office employees and
the cost of mistakes in processing transactions, it might well make sense
to do a cost-benefit analysis of changing the educational requirements for
a back-office job in some of the locations. In some cases, a detailed
analysis of the cause of losses may provide insights. For example, if
40% of computer failures can be attributed to the fact that the current
hardware is several years old and less reliable than newer versions, a costbenefit analysis of upgrading is likely to be useful.
RCSA and KRIs
Risk and control self assessment (RCSA) is an important way in which
banks try and achieve a better understanding of their operational risk
exposures. This involves asking the managers of the business units
themselves to identify their operational risks. Sometimes questionnaires
designed by senior management are used.
A by-product of any program to measure and understand operational
risk is likely to be the development of key risk indicators (KRIs). Risk
indicators are key tools in the management of operational risk. The most
important indicators are prospective. They provide an early-warning
system to track the level of operational risk in the organization. Examples
of key risk indicators are staff turnover and number of failed transactions.
The hope is that key risk indicators can identify potential problems and
allow remedial action to be taken before losses are incurred.
It is important for a bank to quantify operational risks, but it is even
more important to take action to control and manage those risks.
14.6 ALLOCATION OF OPERATIONAL RISK CAPITAL
Operational risk capital should be allocated to business units in a way
that encourages them to improve their operational risk management. If a
business unit can show that it has taken steps to reduce the frequency or
severity of a particular risk, it should be allocated less capital. This will
have the effect of improving the business unit's return on capital (and
possibly lead to the business unit manager receiving an increased
bonus).
Note that it is not always optimal for a manager to reduce a
particular operational risk. Sometimes the costs of reducing the risk
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Chapter 14
outweigh the benefits of reduced capital, so that return on allocated
capital decreases. A business unit should be encouraged to make appropriate calculations and determine the amount of operational risk that
maximizes return on capital.
Scorecard Approaches
Some banks use Scorecard approaches to allocate operational risk capital.
Experts identify the key determinants of each type of risk and then
formulate questions for managers of business units to enable risk levels
to be quantified. The total number of different business units is likely to
be greater than the eight listed in Table 14.1 because each region of the
world in which the bank operates often has to be considered separately.
Examples of the questions that might be used are:
• What is the number of sensitive positions filled by temps?
• What is the ratio of supervisors to staff?
• Does your business have confidential client information?
• What is the employee turnover rate per annum?
• How many open employee positions are there at any time?
• What percentage of your staff has a performance-based component to
their remuneration?
• What percentage of your staff did not take ten consecutive days leave in
the last 12 months?
Scores are assigned to the answers. The total score for a particular
business unit indicates the amount of risk present in the business unit
and can be used as a basis for allocating capital to the business unit. The
scores given by a Scorecard approach should be validated by comparing
scores with actual loss experience whenever possible.
The overall result of operational risk assessment and operational risk
capital allocation should be that business units become more sensitive to
the need for managing operational risk. Hopefully operational risk
management will be seen to be an important part of every manager's
job. A key ingredient for the success of any operational risk program is
the support of senior management. The Basel Committee on Banking
Supervision is very much aware of this. It recommends that the bank
board of directors be involved in the approval of a risk management
program and that it reviews the program on a regular basis.
Operational
Risk
335
14.7 USE OF THE POWER LAW
In Section 5.4 we introduced the power law. This states that for a wide
range of variables
where v is the value of the variable, x is a relatively large value of and K
and are constants. We covered the theoretical underpinnings of the
power law and maximum-likelihood estimation procedures when we
looked at extreme value theory in Section 9.4.
De Fountnouvelle et al, using data on losses from external vendors,
find that the power law holds well for the large losses experienced by
banks.6 This makes the calculation of VaR with high degrees of confidence
such as 99.9% possible. Loss data (internal or external) is used to estimate
the power law parameters using the maximum-likelihood approach in
Chapter 9. The 99.9% quantile of the loss distribution is then estimated
using equation (9.6).
When loss distributions are aggregated, the distribution with the
heaviest tails tends to dominate. This means that the loss with the lowest
defines the extreme tails of the total loss distribution.7 Therefore, if all
we are interested in is calculating the extreme tail of the total operational
risk loss distribution, it may only be necessary to consider one or two
business-line/loss-type combinations.
14.8 INSURANCE
An important decision for operational risk managers is the extent to
which operational risks should be insured against. Insurance policies
are available on many different kinds of risk ranging from fire losses to
rogue trader losses. Provided that the insurance company's balance sheet
satisfies certain criteria, a bank using AMA can reduce the capital it is
required to hold by entering into insurance contracts. In this section we
review some of the key issues facing insurance companies in the design of
their insurance contracts and show how these are likely to influence the
type of contracts that banks can negotiate.
6
See p. De Fountnouvelle, V. DeJesus-Rueff, J. Jordan, and E. Rosengren, "Capital and
Risk: New Evidence on Implications of Large Operational Risk Losses," Federal Reserve
Board of Boston, Working Paper, September 2003.
7
In Chapter 9 the parameter equals
so it is the loss distribution with the largest
that defines the extreme tails.
336
Chapter 14
Moral Hazard
One of the risks facing an insurance company is moral hazard. This is the
risk that the existence of the insurance contract will cause the bank to
behave differently than it otherwise would. This changed behavior increases the risks to the insurance company. Consider, for example, a bank
that insures itself against robberies. As a result of the insurance policy, it
may be tempted to be lax in its implementation of security measures—
making a robbery more likely than it would otherwise have been.
Insurance companies have traditionally dealt with moral hazard in a
number of ways. Typically there is a deductible in any insurance policy.
This means that the bank is responsible for bearing the first part of any
loss. Sometimes there is a coinsurance provision in a policy. The insurance
company then pays a predetermined percentage (less than 100%) of losses
in excess of the deductible. In addition, there is nearly always a policy limit.
This is a limit on the total liability of the insurer. Consider again a bank
that has insured itself against robberies. The existence of deductibles,
coinsurance provisions, and policy limits are likely to provide an incentive
for a bank not to relax security measures in its branches. The moral hazard
problem in rogue trader insurance in discussed in Business Snapshot 14.2.
Adverse Selection
The other major problem facing insurance companies is adverse selection.
This is where an insurance company cannot distinguish between good
and bad risks. It offers the same price to everyone and inadvertently
attracts more of the bad risks. For example, banks without good internal
controls are more likely to enter into rogue trader insurance contracts;
banks without good internal controls are more likely to buy insurance
policies to protect themselves against external fraud.
To overcome the adverse selection problem, an insurance company must
try to understand the controls that exist within banks and the losses that
have been experienced. As a result of its initial assessment of risks, it may
not charge the same premium for the same contract to all banks. Over time
it gains more information about the bank's operational risk losses and may
increase or reduce the premium charged. This is much the same as the
approach adopted by insurance companies when they sell automobile
insurance to a driver. At the outset the insurance company obtains as
much information on the driver as possible. As time goes by, it collects
more information on the driver's risk (number of accidents, number of
speeding tickets, etc.) and modifies the premium charged accordingly.
Operational
337
Risk
Business Snapshot 14.2
Rogue Trader Insurance
A rogue trader insurance policy presents particularly tricky moral hazard
problems. An unscrupulous bank could enter into an insurance contract to
protect itself against losses from rogue trader risk and then choose to be lax in
jits implementation of trading limits. If a trader exceeds the trading limit and
makes a large profit, the bank is better off than it would be otherwise. If a
large loss results, a claim can be made under the rogue trader insurance policy.
Deductibles, coinsurance provisions, and policy limits may mean that the
amount recovered is less than the loss incurred by the trader. However,
potential net losses to the bank are likely to be far less than potential profits,
making the lax trading limits strategy a good bet for the bank.
Given this problem, it is perhaps surprising that some insurance companies
do offer rogue trader insurance policies. These companies tend to specify
carefully how trading limits are implemented. They may also require that the
existence of the insurance policy not be revealed to anyone on the trading floor.
They are likely to want to retain the right to investigate the circumstances
underlying any loss. It is also worth pointing out that, from the bank's point of
view, the lax trading limits strategy we have outlined may be very shortsighted.
The bank might well find that future insurance costs rise significantly as a result
of a rogue trader claim. Furthermore, a large rogue trader loss (even if insured)
would cause its reputation to suffer.
14.9 SARBANES-OXLEY
Largely as a result of the Enron bankruptcy the Sarbanes-Oxley Act was
Passed in the United States in 2002. This provides another dimension to
operational risk management for financial and nonfinancial institutions
in the United States. The Act requires boards of directors to become
mtuch more involved with day-to-day operations. They must monitor
internal controls to ensure risks are being assessed and handled well.
The Act specifies rules concerning the composition of the board of
directors of public companies and lists the responsibilities of the board.
It gives the SEC the power to censure the board or give it additional
responsibilities. A company's auditors are not allowed to carry out any
significant nonauditing services for the company.8 Audit partners must be
rotated. The audit committee of the board must be made aware of
alternative accounting treatments. The CEO and CFO must prepare a
8
Enron's auditor, Arthur Andersen, provided a wide range of services in addition to
auditing. It did not survive the litigation that followed the downfall of Enron.
338
Chapter 14
statement to accompany the audit report to the effect that the financial
statements are accurate. The CEO and CFO are required to return bonuses
in the event that financial statements are restated. Other rules concern
insider trading, disclosure, personal loans to executives, reporting of transactions by directors, and the monitoring of internal controls by directors.
SUMMARY
In 1999, bank supervisors indicated their intention to charge capital for
operational risk. This has led banks to carefully consider how they should
measure and manage operational risk. Bank supervisors have identified
seven different types of operational risk and eight different business lines.
They encourage banks to quantify risks for each of the 56 risk-type/
business-line combinations.
One approach that has been developed is the statistical approach. This
treats operational risk losses in much the same way as actuaries treat losses
from insurance policies. A frequency of loss distribution and a severity of
loss distribution is estimated and these are combined to form a total
operational loss distribution. If possible, the frequency of loss distribution
is estimated from internal data. The loss severity distribution is estimated
from a combination of internal and external data.
There are two sources of external data. One is data obtained from other
banks via sharing arrangements; the other is publicly available data on
large losses collected by data vendors. Increasingly banks are augmenting
loss data with scenario analyses where senior managers develop loss-event
scenarios and estimate parameters describing loss frequency and severity.
Risk managers should try to be forward-looking in their approach to
operational risk. They should try to understand what determines operational risk losses and develop key risk indicators to track the level of
operational risk in different parts of the organization.
Once operational risk capital has been estimated, it is important to
develop procedures for allocating it to business units. This should be done
in a way that encourages business units to reduce operational risk when
they can do so without incurring excessive costs. One approach to
allocation is the use of scorecards.
The power law introduced in Chapter 5 seems to apply to operational
risk losses. This makes it possible to use extreme value theory to estimate
the tails of a loss distribution from empirical data. When several loss
distributions are aggregated, it is the loss distribution with the heaviest
Operational Risk
339
tail that dominates. In principle, this makes the calculation of VaR for
total operational risk easier.
Many operational risks can be insured against. However, most policies
include deductibles, coinsurance provisions, and policy limits. As a result
a bank is always left bearing part of any risk itself. Moreover, the way
insurance premiums change as time passes is likely to depend on the
claims made and other indicators that the insurance company has of how
well operational risks are being managed.
The whole process of measuring, managing, and allocating operational
risk is still in its infancy. As time goes by and data is accumulated, more
precise procedures than those we have mentioned in this chapter are likely
to emerge. One of the key problems is that there are two sorts of
operational risk: high-frequency low-severity risks and low-frequency
high-severity risks. The former are relatively easy to quantify, but operational risk VaR is largely driven by the latter.
Bank supervisors seem to be succeeding in their objective of making
banks more sensitive to the importance of operational risk. In many ways
the key benefit of an operational risk management program is not the
numbers that are produced, but the process that banks go through in
producing the numbers. If well handled, the process can sensitize managers to the importance of operational risk and perhaps lead to them
thinking about it differently.
FURTHER READING
Bank for International Settlements, "Sound Practices for the Management and
Supervision of Operational Risk," February 2003.
Baud, N., A. Frachot, and T. Roncalli, "Internal Data, External Data and
Consortium Data for Operational Risk Management: How to Pool Data
Properly," Working Paper, Groupe de Recherche Operationelle, Credit
Lyonnais, 2002.
Chorafas, D. N., Operational Risk Control with Basel II: Basic Principles and
Capital Requirements. Elsevier, 2003.
De Fountnouvelle, P., V. DeJesus-Rueff, J. Jordan, and E. Rosengren, "Capital
and Risk: New Evidence on Implications of Large Operational Risk Losses,"
Federal Reserve Board of Boston, Working Paper, September 2003.
Netter, J., and A. Poulsen, "Operational Risk in Financial Service Providers and
the Proposed Basel Accord: An Overview," Working Paper, Terry College of
Business, University of Georgia.
340
Chapter 14
Van Den Brink, G. J., Operational Risk: The New Challenge for Banks.
Basingstoke, UK: Palgrave, 2001.
QUESTIONS AND PROBLEMS (Answers at End of Book)
14.1. What risks are included by regulators in their definition of operational
risks? What risks are not included?
14.2. Suppose that external data shows that a loss of $100 million occurred at a
bank with annual revenues of $1 billion. Your bank has annual revenues
of $3 billion. What is the implication of the external data for losses that
could occur at your bank.
14.3. Suppose that there is a 90% probability that operational risk losses of a
certain type will not exceed $20 million. The power law parameter is 0.8.
What is the probability of losses exceeding (a) $40 million, (b) $80 million,
and (c) $200 million.
14.4. Discuss how moral hazard and adverse selection are handled in car
insurance.
14.5. Give two ways Sarbanes-Oxley affects the CEOs of public companies.
14.6. When is a trading loss classified as a market risk and when is it classified
as an operational risk?
14.7. Discuss whether there is (a) moral hazard and (b) adverse selection in life
insurance contracts.
14.8. What is external loss data? How is it obtained? How is it used in
determining operational risk loss distributions for a bank?
14.9. What distributions are commonly used for loss frequency and loss
severity?
14.10. Give examples of key risk indicators that might be monitored by a central
operational risk management group within a bank.
14.11. The worksheet used to produce Figure 14.2 is on the author's website
What is the mean and standard deviation of the loss distribution. Modify
the inputs to the simulation to test the effect of changing the loss
frequency from 3 to 4.
ASSIGNMENT QUESTIONS
14.12. Suppose that there is a 95% probability that operational risk losses of a
certain type exceed $10 million. Use the power law to estimate the
99.97% worst-case operational risk loss when the
parameter equals
(a) 0.25, (b) 0.5, (c) 0.9, and (d) 1.0. ,.
Operational
Risk
341
14.13. Consider the following two events: (a) a bank loses $1 billion from an
unexpected lawsuit relating to its transactions with a counterparty and
(b) an insurance company loses $1 billion because of an unexpected
hurricane in Texas. Suppose you own shares in both the bank and the
insurance company. Which loss are you more concerned about? Why?
14.14. The worksheet used to produce Figure 14.2 is on the author's website.
How does the loss distribution change when the loss severity has a beta
distribution with an upper bound of 5, a lower bound of 0, and the other
parameters both 1?
Model Risk and
Liquidity Risk
In this chapter we discuss two additional types of risk faced by financial
institutions: model risk and liquidity risk. Model risk is the risk related to
the models a financial institution uses to value derivatives. Liquidity risk
is the risk that there may not be enough buyers (or sellers) in the market
for a financial institution to execute the trades it desires. The two risks are
related. Sophisticated models are only necessary to price products that are
relatively illiquid. When there is an active market for a product, prices can
be observed in the market and models play a less important role.
There are two main types of model risk. One is the risk that the model
will give the wrong price at the time a product is bought or sold. This can
result in a company buying a product for a price that is too high or selling
it for a price that is too low. The other risk concerns hedging. If a
company uses the wrong model, the Greek letters it calculates—and the
hedges it sets up based on those Greek letters—are liable to be wrong.
Liquidity risk is the risk that, even if a financial institution's theoretical
Price is in line with the market price and the price of its competitors, it
cannot trade in the volume required at the price. Suppose that the offer
Price for a particular option is $40. The financial institution could
Probably buy 10,000 options at this price. But it is likely to be quite
difficult to buy 10 million options at or close to the price. If the financial
institution went into the market and started buying large numbers of
options from different market makers, then the price of the option would
Probably go up, making the rest of its trades more expensive.
344
Chapter 15
15.1 THE NATURE OF MODELS IN FINANCE
Many physicists work in the front and middle office of banks and many of
the models they use are similar to those encountered in physics. For
example, the differential equation that leads to the famous Black-Scholes
model is the heat-exchange equation that has been used by physicists for
many years. However, as Derman has pointed out, there is an important
difference between the models of physics and those of finance.1 The models
of physics describe physical processes and are highly accurate. By contrast,
the models of finance describe the behavior of market variables. This
behavior depends on the actions of human beings. As a result the models
are at best approximate descriptions of the market variables. This is why
the use of models in finance entails what is referred to as "model risk".
One important difference between the models of physics and the
models of finance concerns model parameters. The parameters of models
in the physical sciences are usually constants that do not change. The
parameters in finance models are often assumed to be constant for the
whole life of the model when the model is used to calculate an option
price on any particular day. But the parameters are changed from day to
day so that market prices are matched. The process of choosing model
parameters is known as calibration.
An example of calibration is the choice of the volatility parameter in the
Black-Scholes model. This model assumes that volatility remains constant
for the life of the model. However, the volatility parameter that is used in
the model changes daily. For a particular option maturing in three months,
the volatility parameter might be 20% when the option is valued today,
22% when valued tomorrow, and 19% when valued on the next day. For
some models in finance, the calibration process is quite involved. For
example, calibrating an interest rate model on a particular day involves
(a) fitting the zero-coupon yield curve observed on that day and (b) fitting
the market prices of actively traded interest rate options such as caps and
swap options.
Sometimes parameters in finance models have to be calibrated to
historical data rather than to market prices. Consider a model involving
both an exchange rate and an equity index. It is likely that the correlation
between the exchange rate movements and the equity price movements
would be estimated from historical data because there are no actively
traded instruments from which the correlation can be implied.
1
See E. D e r m a n , My Life as a Quant: Reflections on Physics and Finance, Wiley, 2004
Model Risk and Liquidity Risk
Business Snapshot 15.1
345
Kidder Peabody's Embarrassing Mistake
Investment banks have developed a way of creating a zero-coupon bond, called
a strip, from a coupon-bearing Treasury bond by selling each of the cash flows
underlying the coupon-bearing bond as a separate security. Joseph Jett, a
trader working for Kidder Peabody, had a relatively simple trading strategy.
He would buy strips and sell them in the forward market. The forward price of
the strip was always greater than the spot price and so it appeared that he had
found a money machine! In fact the difference between the forward price and
the spot price represents nothing more than the cost of funding the purchase
of the strip. Suppose, for example, that the three-month interest rate is 4% per
annum and the spot price of a strip is $70. The three-month forward price of
the strip is 70e 0.04x3/12 = $70.70.
Kidder Peabody's computer system reported a profit on each of Jett's trades
equal to the excess of the forward price over the spot price ($0.70 in our
example). By rolling his contracts forward, Jett was able to prevent the
funding cost from accruing to him. The result was that the system reported
a profit of $100 million on Jett's trading (and Jett received a big bonus) when
in fact there was a loss in the region of $350 million. This shows that even large
financial institutions can get relatively simple things wrong!
15.2 MODELS FOR LINEAR PRODUCTS
Pricing linear products such as forward contracts and swaps is straightforward and relies on little more than present value arithmetic. There is
usually very little disagreement in the market on the correct pricing models
for these products and very little model risk. However, this does not mean
that there is no model risk. As indicated in Business Snapshot 15.1, Kidder
Peabody's computer system did not account correctly for funding costs
when a linear product was traded. As a result the system indicated that one
of the company's traders was making a large profit when in fact he was
making a huge loss.
Another type of model risk arises when a financial institution assumes
a product is simpler than it actually is. Consider the interest rate swap
market. A plain vanilla interest rate swap such as that described in
Section 2.3 can be valued by assuming that forward interest rates will
be realized as described in Appendix B. For example, if the forward
interest rate for the period between 2 and 2.5 years is 4.3%, we value the
swap on the assumption that the floating rate that is exchanged for fixed
at the 2.5-year point is 4.3%.
346
Chapter 15
Business Snapshot 15.2 Exploiting the Weaknesses of a
Competitor's Model
A LIBOR-in-arrears swap is an interest rate swap where the floating interest
rate is paid on the day it is observed, not one accrual period later. Whereas a
plain vanilla swap is correctly valued by assuming that future rates will be
today's forward rates, a LIBOR-in-arrears swap should be valued on the
assumption that the future rate is today's forward interest rate plus a "convexity
adjustment".
In the mid-1990s sophisticated financial institutions understood the correct
approach for valuing a LIBOR-in-arrears swap. Less sophisticated financial
institutions used the naive "assume forward rates will be realized" approach
The result was that by choosing trades judiciously sophisticated financial
institutions were able to make substantial profits at the expense of their less
sophisticated counterparts.
The derivatives business is one where traders do not hesitate to exploit the
weaknesses of their competitor's models!
It is tempting to generalize from this and argue that any agreement to
exchange cash flows can be valued on the assumption that forward rates
are realized. This is not so. Consider, for example, what is known as a
LIBOR-in-arrears swap. In this instrument the floating rate that is observed
on a particular date is paid on that date (not one accrual period later as is
the case for a plain vanilla swap). A LIBOR-in-arrears swap should be
valued on the assumption that the realized interest rate equals the forward
interest rate plus a "convexity adjustment". As indicated in Business
Snapshot 15.2, financial institutions that did not understand this lost
money in the mid-1990s.
15.3 MODELS FOR ACTIVELY TRADED PRODUCTS
When a product trades actively in the market, we do not need a model to
know what its price is. The market tells us this. Suppose, for example,
that a certain option on a stock index trades actively and is quoted by
market makers as bid $30 and offer $31. Our best estimate of its current
value is the mid-market price of $30.50.
A model is often used as a communication tool in these circumstances.
Traders like to use models where only one of the variables necessary to
determine the price of a product is not directly observable in the market.
The model then provides a one-to-one mapping of the product's price to
Model Risk
and Liquidity
Risk
347
the variable and vice versa. The Black-Scholes model (see Appendix C) is a
case in point. The only unobservable variable in the model is the volatility
of the underlying asset. The model therefore provides a one-to-one mapping of option prices to volatilities and vice versa. As explained in Chapter
5, the volatility calculated from the market price is known as the implied
volatility. Traders frequently quote implied volatilities rather than the
dollar prices. The reason is that the implied volatility is more stable than
the price. For example, when the underlying asset price or the interest rate
changes, there is likely to be a much bigger percentage jump in the dollar
price of an option than in its implied volatility.
Consider again the index option that has a mid-market price of
$30.50. Suppose it is a one-year European call option where the strike
price is 1,000, the one-year forward price of the index is 1,100, and the
one-year risk-free interest rate is 3%. The mid-market implied volatility
would be quoted as 15.37%.2 The bid-offer spread might be "bid
15.24%, offer 15.50%".
Volatility Smiles
The volatility implied by Black-Scholes (or by a binomial tree calculation such as that in Appendix D) as a function of the strike price for a
particular option maturity is known as a volatility smile.3 If traders really
believed the assumptions underlying the Black-Scholes model, the
Figure 15.1
Volatility smile for foreign currency options.
2
Implied volatility calculations can be done with the DerivaGem software available on
the author's website.
3
It can be shown that the relationship between strike price and implied volatility should
be exactly the same for calls and puts in the case of European options and approximately
the same in the case of American options.
348
Chapter 15
Figure 15.2 Volatility smile for equity options.
implied volatility would be the same for all options and the volatility
smile would be flat. In fact, this is rarely the case.
The volatility smile used by traders to price foreign currency options
has the general form shown in Figure 15.1. The volatility is relatively low
for at-the-money options. It becomes progressively higher as an option
moves either in the money or out of the money. The reason for the
volatility smile is that Black-Scholes assumes
1. The volatility of the asset is constant.
2. The price of the asset changes smoothly with no jumps.
In practice, neither of these conditions is satisfied for an exchange rate. The
volatility of an exchange rate is far from constant, and exchange rates
frequently exhibit jumps. 4 It turns out that the effect of both a nonconstant
volatility and jumps is that extreme outcomes become more likely. This
leads to the volatility smile in Figure 15.1.
The volatility smile used by traders to price equity options (both those
on individual stocks and those on stock indices) has the general form
shown in Figure 15.2. This is sometimes referred to as a volatility skew. The
4
Often the jumps are in response to the actions of central banks.
Model Risk and Liquidity Risk
Business Snapshot 15.3
349
Crashophobia
It is interesting that the pattern in Figure 15.2 for equities has existed only
since the stock market crash of October 1987. Prior to October 1987 implied
volatilities were much less dependent on strike price. This has led Mark
Rubinstein to suggest that one reason for the equity volatility smile may be
"Crashophobia". Traders are concerned about the possibility of another crash
similar to October 1987 and assign relatively high prices (and therefore
relatively high implied volatilities) for deep-out-of-the-money puts.
There is some empirical support for this explanation. Declines in the
S&P 500 tend to be accompanied by a steepening of the volatility skew,
perhaps because traders become more nervous about the possibility of a crash.
When the S&P increases, the skew tends to become less steep.
volatility decreases as the strike price increases. The volatility used to price
a low-strike-price option (i.e., a deep-out-of-the-money put or a deep-inthe-money call) is significantly higher than that used to price a high-strikeprice option (i.e., a deep-in-the-money put or a deep-out-of-the-money
call). One possible explanation for the smile in equity options concerns
leverage. As a company's equity declines in value, the company's leverage
increases. This means that the equity becomes more risky and its volatility
increases. As a company's equity increases in value, leverage decreases.
The equity then becomes less risky and its volatility decreases. This
argument shows that we can expect the volatility of equity to be a
decreasing function of price and is consistent with Figure 15.2. Another
explanation is Crashophobia (see Business Snapshot 15.3).
Volatility smiles and skews such as those shown in Figures 15.1 and 15.2
are liable to change on a daily basis. This means that the volatilities
traders use change from day to day as well as from option to option. Why
does the market continue to use Black-Scholes (and its extensions) when
it provides such a poor fit to market prices? The answer is that traders like
the model. They find it easy to use and easy to understand.
Volatility Surfaces
Figures 15.1 and 15.2 are for options with a particular maturity. Traders
like to combine the volatility smiles for different maturities into a volatility
surface. This shows implied volatility as a function of both strike price
and time to maturity. Table 15.1 gives a typical volatility surface for
currency options. The table indicates that the volatility smile becomes less
350
Chapter 15
Table 15.1
Volatility surface.
Time to
maturity
0.90
0.95
1.00
1.05
1.10
1
3
6
1
2
5
14.2
14.0
14.1
14.7
15.0
14.8
13.0
13.0
13.3
14.0
14.4
14.6
12.0
12.0
12.5
13.5
14.0
14.4
13.1
13.1
13.4
14.0
14.5
14.7
14.5
14.2
14.3
14.8
15.1
15.0
month
month
month
year
year
year
Strike price
pronounced as the time to maturity increases. This is what is usually
observed in practice.5
The volatility surface is produced primarily from information provided
by brokers. Brokers are in the business of bringing buyers and sellers
together in the over-the-counter market and have more information on
the implied volatilities at which transactions are being done on any given
day than individual derivatives dealers. Over time an options trader
develops an understanding of what the volatility surface for a particular
underlying asset should look like.
To value a new option, traders look up the appropriate volatility in the
table using interpolation. For example, to value a 9-month option with a
strike price of 1.05, a trader would interpolate between 13.4 and 14.0 in
Table 15.1 to obtain a volatility of 13.7%. This is the volatility that would
be used in a Black-Scholes formula or a binomial tree calculation. When
valuing a 1.5-year option with a strike price of 0.925, a two-dimensional
interpolation would be used to give an implied volatility of 14.525%.
Hedging
It should be clear from the above discussion that models play a relatively
minor role in the pricing of actively traded products. Dealers interpolate
between prices observed in the market. A model such as Black-Scholes is
nothing more than a tool to facilitate the interpolation. It is easier to
interpolate between implied volatilities than between dollar prices.
5
If T is the time to maturity and F0 is the forward price of the asset, some traders choose
to define the volatility smile as the relationship between implied volatility and
rather than as the relationship between the implied volatility and K. The smile is
usually much less dependent on the time to maturity.
Model
Risk and Liquidity Risk
351
Models are used in a more significant way when it comes to hedging.
Traders must manage their exposure to delta, gamma, vega, etc. (see
Chapter 3). We can distinguish between within-model hedging and
outside-model hedging. Within-model hedging is designed to deal with
the risk of changes in variables that are assumed to be uncertain by the
model. Outside-model hedging deals with the risk of changes in variables
that are assumed to be constant (or deterministic) by the model. When
Black-Scholes is used, hedging against movements in the underlying stock
price (delta and gamma hedging) is within-model hedging because the
model assumes that stock price changes are uncertain. However, hedging
against volatility (vega hedging) is outside-model hedging because the
model assumes that volatility is constant.
In practice, traders almost invariably do outside-model hedging as well
as within-model hedging. This is because, as we have explained, the
calibration process causes parameters such as volatilities to change daily.
A natural assumption is that if hedging is implemented for all the
variables that could change in a day (both those that are assumed to
be constant by the model and those that are assumed to be stochastic) the
value of hedger's position will not change. In fact, this is not necessarily
the case. If the model used to calculate the hedge is wrong, then there may
be an unexpected gain or loss. The good news here is that on average the
gain or loss from hedging using the wrong model is approximately zero.
The risk of imperfect hedging is likely to be largely diversified away across
the portfolio of a large financial institution.
Many financial institutions carefully evaluate the effectiveness of their
hedging. They find it revealing to decompose the day-to-day change in a
portfolio's value into the following:
1. A change resulting from risks that were unhedged
2. A change resulting from the hedging model being imperfect
3. A change resulting from new trades done during the day
This is sometimes referred to as a P&L decomposition.
15.4 MODELS FOR STRUCTURED PRODUCTS
Exotic options and other nonstandard products that are tailored to the
needs of clients are referred to as structured products. Usually they do not
trade actively and a financial institution must rely on a model to determine the price it charges the client. Note the important difference between
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structured products and actively traded products. When a product trades
actively, there is very little uncertainty about its price and the model
affects only hedge performance. In the case of structured products, model
risk is much greater because there is the potential for both pricing and
hedging being incorrect.
A financial institution should not rely on a single model for pricing
structured products. Instead it should, whenever possible, use several
different models. This leads to a price range for the instrument and a
better understanding of the model risks being taken.
Suppose that three different models give prices of $6 million, $7.5 million
and $8.5 million for a particular product that a financial institution is
planning to sell to a client. Even if the financial institution believes that the
first model is the best one and plans to use that model as its standard model
for daily repricing and hedging, it should ensure that the price it charges
the client is at least $8.5 million. Moreover, it should be conservative about
recognizing profits. If the product is sold for $9 million, it is tempting to
recognize an immediate profit of $3 million ($9 million less the believed-tobe-accurate price of $6 million). However, this is overly aggressive. A
better, more conservative, practice is to put the $3 million into a reserve
account and transfer it to profits slowly during the life of the product.6
Most large financial institutions have model audit groups as part of
their risk management teams. These groups are responsible for vetting
new models proposed by traders for particular products. A model cannot
usually be used until the model audit group has approved it. Vetting
typically includes (a) checking that a model has been correctly implemented, (b) examining whether there is a sound rationale for the model,
(c) comparing the model with other models that can accomplish the same
task, (d) specifying the limitations of the model, and (e) assessing
uncertainties in the prices and hedge parameters given by the model.
15.5 DANGERS IN MODEL BUILDING
The art of model building is to capture what is important for valuing
and hedging an instrument without making the model more complex
than it needs to be. Sometimes models have to be quite complex to
capture the important features of a product, but this is not always
the case.
6
This is also likely to have sensible implications for the way bonuses are paid.
Model Risk and Liquidity Risk
353
One danger in model building is Overfitting. Consider the problem
posed by the volatility surface in Table 15.1. We can exactly match the
volatility surface with a single model by extending Black-Scholes so that
volatility is a complex function of the underlying asset price and time.7
But when we do this, we may find that other properties of the model are
less reasonable than those of simpler models. In particular, the joint
probability of the asset prices at two or more times may be unrealistic.8
Another danger in model building is Overparameterization. The BlackScholes model can be extended to include features such as a stochastic
volatility or jumps in the asset price. This invariably introduces extra
parameters that have to be estimated. It is usually claimed that the
parameters in complex models are more stable those in simpler models
and do not have to be adjusted as much from day to day. This may be
true, but we should remember that we are not dealing with physical
processes. The parameters in a complex model may remain relatively
constant for a period of time and then change, perhaps because there
has been what economists refer to as a regime shift. A financial institution
may find that a complicated model is an improvement over a simple
model until the parameters change. At that time it may not have the
flexibility to cope with changing market conditions.
As we have mentioned, traders like simple models that have just one
unobservable parameter. They are skeptical of more complex models
because they are "black boxes" and it is very difficult to develop intuition
about them. In some situations their skepticism is well founded for the
reasons we have just mentioned.
15.6 DETECTING MODEL PROBLEMS
The risk management function within a financial institution should carefully monitor the financial institution's trading patterns. In particular it
7
This is the implied volatility function model proposed by B. Dupire, "Pricing with a
Smile," Risk, 7 (February 1994), 18-20; E. Derman and I. Kani, "Riding on a Smile,"
Risk, 7 (February 1994), 32-39; M. Rubinstein, "Implied Binomial Trees," Journal of
Finance, 49, 3 (July 1994), 771-818.
8
Instruments such as barrier options and compound options depend on the joint
Probability distribution of the asset price at different times. Hull and Suo find that the
implied volatility function model works reasonably well for compound options, but
sometimes gives serious errors for barrier options. See J. C. Hull and W. Suo, "A
Methodology for the Assessment of Model Risk and its Application to the Implied
Volatility Function Model," Journal of Financial and Quantitative Analysis, 37, 2 (June
2002), 297-318.
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should keep track of the following:
1. The type of trading the financial institution is doing with other
financial institutions
2. How competitive it is in bidding for different types of structured
transactions
3. The profits being recorded from the trading of different products
Getting too much of a certain type of business or making huge profits
from relatively simple trading strategies can be a warning sign. Another
clear indication that something is wrong is when the financial institution
is unable to unwind trades at close to the prices given by its computer
models.
The high profits being recorded for Joseph Jett's trading at Kidder
Peabody (see Business Snapshot 15.1) should have been a warning sign.9
Furthermore, if in the mid-1990s a financial institution's risk management team discovered that traders were entering into a large number of
LIBOR-in-arrears swaps with other financial institutions (see Business
Snapshot 15.2) where they were receiving fixed and paying floating, they
could have alerted modelers to a potential problem and directed that
trading in the product be temporarily stopped.
There are other ways in which a derivatives dealer might find that one of
its models is out of line with that used by other market participants.
Dealers often subscribe to services that are designed to provide market
quotes for representative trades. Typically the company providing this
service periodically asks its dealer clients for quotes on specific hypothetical transactions. It then averages the quotes (possibly after eliminating
the highest and lowest) and feeds the results back to the dealers.
15.7 TRADITIONAL VIEW OF LIQUIDITY RISK
Liquidity is liable to affect both the funding and trading activities of a
financial institution. We start by considering trading activities and move
on to consider funding activities in Section 15.10.
The traditional view of liquidity risk in trading is that there is a
relationship between price and quantity. This relationship is shown in
Figure 15.3. When the quantity of an asset that is traded is relatively
9
Barry Finer, risk manager for the government bond desk at Kidder Peabody, did point
out the difficulty of making large arbitrage profits from a market as efficient as the US
government bond market, but his concerns were dismissed out of hand.
Model Risk and Liquidity Risk
Figure 15.3
355
Bid and offer prices as a function of quantity
transacted.
small, bid-offer spreads are low. As the quantity increases, the price paid
by the buyer of the asset increases and the price received by the seller of
the asset decreases.
How can a financial institution manage liquidity in the trading book?
One way is by using position limits. If the size of the financial institution's
position is limited, the size of a trade it has to do to unwind a position is
also limited. It is often argued that the time horizon in a VaR calculation
should reflect the time necessary to unwind a position. If a position can
be unwound very quickly, a one-day time horizon is appropriate; in other
circumstances, time horizons as long as one month may be needed.
Liquidity-Adjusted VaR
The percentage bid-offer spread for an asset can be defined as
where the mid-price is halfway between the bid and the offer. In liquidating a position in the asset, a financial institution incurs a cost equal to
where is the dollar value of the position. This reflects the fact that
trades are not done at the mid-market price. A buy trade is done at a
Proportional amount s/2 above the market price and a sell trade is done
at a proportional amount s/2 below the market price.
Risk managers sometimes calculate a liquidity-adjusted VaR by adding
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for each position in the book. Formally, we have
where n is the number of positions, is the percentage bid-offer spread for
the ith position and is the amount of money invested in the ith position.
As the number of positions, n, grows, VaR benefits from diversification but
the liquidity adjustment does not. Consequently, the percentage difference
between VaR and liquidity-adjusted VaR grows as n grows.
A variation on this calculation that takes account of uncertainty in the
spread has been suggested by Bangia et al.10 This involves estimating the
mean
and standard deviation of and defining
The parameter gives the required confidence level for the spread on the
assumption that spreads are normally distributed. For example, if a 95%
confidence level is required, then = 1.64. Bangia et al.'s equation
assumes (conservatively) that spreads in all instruments are perfectly
correlated.
15.8 LIQUIDITY BLACK HOLES
The liquidity risk we have just described is real. Banks cannot trade at
mid-market prices and the bigger the size of their transaction the higher
the bid-offer spread that they face. However, there is a more serious
liquidity risk. This is the risk that liquidity in a particular market will dry
up completely because everyone wants to buy and no-one wants to sell, or
vice versa.
It is sometimes argued that technological and other developments have
led to a steady improvement in the liquidity of financial markets. This is
questionable. It is true that bid-offer spreads have on average declined.
However, there has been an increasing tendency toward "herd behavior
where almost everyone wants to do the same type of trade at particular
10
See A. Bangia, F. Diebold, T. Schuermann, and J. Stroughair, "Liquidity on the
Outside," Risk, 12 (June), 68-73.
Model Risk and Liquidity Risk
357
times. The result has been what Persaud refers to as "liquidity black
holes" occurring with increasing frequency.11
In a well-functioning market, the market may change its opinion about
the price of an asset because of new information. However, the price does
not overreact. If a price decrease is too great, traders will quickly move in
and buy the asset and a new equilibrium price will be established. A
liquidity black hole is created when a price decline causes more market
participants to want to sell, driving prices well below where they will
eventually settle. During the sell-off, liquidity dries up and the asset can
be sold only at a fire-sale price.12
Among the reasons for herd behavior and the creation of liquidity
black holes are:
1. The computer models used by different traders are similar.
2. All financial institutions are regulated in the same way and respond
in the same way to changes in volatilities and correlations.
3. There is a natural tendency to feel that if other people are doing a
certain type of trade then they must know something that you do not.
Computer Models
A classic example of computer models causing a liquidity black hole is
the stock market crash of October 1987. In the period leading up to the
crash, the stock market had performed very well. Increasing numbers of
portfolio managers were using commercially available programs to
synthetically create put options on their portfolios. These programs told
them to sell part of their portfolio immediately after a price decline and
buy it back immediately after a price increase. The result, as indicated in
Business Snapshot 15.4, was prices plunging well below their long-run
equilibrium levels on October 19, 1987.
As another example of computer models leading to liquidity black
holes, consider the situation where financial institutions are on one side
of the market for a derivative and their clients are on the other side. When
the price of the underlying asset moves, all financial institutions execute
the same trades to maintain a delta-neutral position. This causes the price
of the asset to move further in the same direction. An example of this is
outlined in Business Snapshot 15.5.
11
See A. D. Persaud (ed.), Liquidity Black Holes: Understanding, Quantifying and
Managing Financial Liquidity Risk, Risk Books, 1999.
12
Liquidity black holes tend to be associated with price decreases, but it is possible for
thern to occur when there are price increases.
\
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Chapter 15
Business Snapshot 15.4
The Crash of 1987
On Monday, October 19, 1987, the Dow J ones Industrial Average dropped by
more than 20%. Portfolio insurance played a major role in this crash. In
October 1987 between $60 billion and $90 billion of equity assets were subject
to portfolio insurance schemes where put options were created synthetical
using a type of "stop-loss" trading strategy.
During the period Wednesday, October 14, 1987, to Friday, October
1987, the market declined by about 10%, with much of this decline take
place on Friday afternoon. The portfolio insurance schemes should have
generated at least $12 billion of equity or index futures sales as a result
this decline. In fact, portfolio insurers had time to sell only $4 billion and then
approached the following week with huge amounts of selling already dictated
by their models. It is estimated that, on Monday, October 19, sell programs
three portfolio insurers accounted for almost 10% of the sales on the New
York Stock Exchange, and that portfolio insurance sales amounted to 21.3%
of all sales in index futures markets. It is likely that the decline in equity pri???
was exacerbated by investors other than portfolio insurers selling heavily
anticipation of the actions of portfolio insurers.
As the market declined so fast and the stock exchange systems were overloaded, many portfolio insurers were unable to execute the trades generated
their models and failed to obtain the protection they required. Needless to say ,
the popularity of portfolio insurance schemes has declined significantly since
1987. One of the morals of this story is that it is dangerous to follow a
particular trading strategy—even a hedging strategy—when many other market participants are doing the same thing.
The Impact of Regulation
In many ways it is a laudable goal on the part of regulators to seek to
ensure that banks and other financial institutions throughout the world
are regulated in the same way. As we explained in Chapter 7, capital
requirements and the extent to which they were enforced varied from
country to country prior to Basel I. Banks were competing globally and
as a result a bank subject to low capital requirements, or capital requirements that were not strictly enforced, had a competitive edge.
However, a uniform regulatory environment comes with costs. All
banks tend to respond in the same way to external events. Consider,
for example, market risk. When volatilities and correlations increase,
market VaR and the capital required for market risks increase. As a
result, banks tend to take steps to reduce their exposures. Since banks
Model
Risk and Liquidity Risk
Business Snapshot 15.5
359
British Insurance Companies
In the late 1990s, British insurance companies had entered into many contracts
promising that the rate of interest applicable to an annuity received by an
individual on retirement would be the greater of the market rate and a
guaranteed rate. At about the same time, largely because of regulatory
pressures, all insurance companies decided to hedge part of their risks on
these contracts by buying long-dated swap options from financial institutions.
The financial institutions they dealt with hedged their risks by buying large
numbers of long-dated sterling bonds. As a result, bond prices rose and
sterling long-term interest rates declined. More bonds had to be bought to
maintain the dynamic hedge, long-term sterling interest rates declined further,
and so on. Financial institutions lost money and, because long-term interest
rates declined, insurance companies found themselves in a worse position on
the risks that they had chosen not to hedge.
often have similar positions to each other, they try to do similar trades. A
liquidity black hole can then develop.
Consider next credit risk. During the low point of the economic cycle,
default probabilities are relatively high and capital requirements for loans
under the Basel II internal ratings based models tend to be high. As a
result banks may be less willing to make loans, creating a liquidity black
hole for small and medium-sized businesses. The Basel Committee has
recognized this as a problem and has dealt with it by asserting that the
probability of default should be an average of the probability of default
through the economic or credit cycle, rather than an estimate applicable
to one particular point in time.
The Importance of Diversity
Economic models usually assume that market participants act independently. We have argued that this is often not the case. It is this lack of
independence that causes liquidity black holes. To solve the problem of
liquidity black holes, we need more diversity in financial markets.
One conclusion from the arguments we have put forward is that a
contrarian investment strategy has some merit. If markets overreact an
investor can do quite well by buying when everyone else is selling and
there is very little liquidity. However, it can be quite difficult for a fund to
follow such a strategy if it is subject to the VaR-based risk management
measures that have become standard.
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Volatilities and correlations tend to be "mean reverting". They sometimes increase but over time they get pulled back to long-run average
levels. One way of creating diversity is to recognize in regulation and in
risk management practices that not all market participants should be
concerned about short-term changes in volatilities and correlations. Asset
managers, for example, should base their decisions on long-term average
volatilities and correlations. They should not join the herd when liquidity
holes develop.
Hedge Funds
Hedge funds have become important participants in financial markets in
recent years. A hedge fund is similar to a mutual fund in that it invests
money on behalf of clients. However, unlike mutual funds hedge funds
are not required to register under US federal securities law. This is
because they accept funds only from financially sophisticated individuals
and do not publicly offer their securities. Mutual funds are subject to
regulations requiring that shares in the funds be fairly priced, that the
shares be redeemable at any time, that investment policies be disclosed,
that the use of leverage be limited, that no short positions be taken, and
so on. Hedge funds are relatively free of these regulations. This gives them
a great deal of freedom to develop sophisticated, unconventional, and
proprietary investment strategies. The fees charged by hedge fund managers are dependent on the fund's performance and are relatively high—
typically 1 to 2% of the amount invested plus 20% of the profits.
Hedge funds have grown in popularity with about $1 trillion being
invested throughout the world for clients in 2004. "Funds of funds" have
been set up to invest in a portfolio of other hedge funds. At the time of
writing, hedge funds are still largely unregulated. This means that they do
not have to assess risk in the same way as other financial institutions. As a
result, hedge funds are in an ideal position to provide liquidity when black
holes show signs of developing. If hedge funds are regulated in the future, it
is to be hoped that the regulations will not be the same as those applying to
other financial institutions.
15.9 LONG-TERM CAPITAL MANAGEMENT
Hedge funds themselves can run into liquidity problems and create or
exacerbate liquidity black holes. The most famous example here is Long-
Model
Risk
and Liquidity
Risk
361
Term Capital Management (LTCM) which was discussed in Business
Snapshot 12.1.
LTCM's problems were exacerbated by the fact that its leverage was
huge. It had about $125 billion of assets (plus large numbers of offbalance-sheet derivatives transactions such as swaps) and only $5 billion
of capital. It was unable to make the payments required under its Collateralization agreements. There was a great deal of concern about the ability of
the financial system to cope with a potential failure of LTCM. What
actually happened was a cash injection by a group of banks and an orderly
liquidation that led to a total loss of about $4 billion. If the fund had been
less highly leveraged, it would probably have been able to survive the flight
to quality and could have waited for the previous relationship between the
prices of the liquid and illiquid securities to resume.
Why was the flight to quality so large? One reason is that there were
rumors in the market that LTCM was experiencing financial difficulties.
These rumors led people to anticipate the sort of trades LTCM would
have to do to close out its positions and the likely effect of those trades on
market prices. When everyone anticipates that something will happen in
financial markets it tends to happen. Another reason is that LTCM had
been highly successful during the 1995 to 1997 period. As a result there
were many other hedge funds trying to imitate its strategy. These hedge
funds also experienced financial difficulties and tried to close out their
positions. This accentuated market movements.
15.10 LIQUIDITY vs. PROFITABILITY
Finally it should be noted that there can be liquidity problems without
profitability problems. For example, a profitable bank can experience a
run on deposits and run into liquidity problems. Banking is to a large
extent about confidence. A bank relies on the withdrawal of deposits
being roughly balanced by new deposits so that funding from liabilities
remains roughly constant (see Section 1.3). If there is a temporary shortfall, it is handled by interbank borrowing.13 However, if there is a loss of
confidence in the bank—however unjustified this might be—the bank is
liable to experience catastrophic liquidity problems.
Liquidity funding problems can be experienced by all sorts of companies. We have all heard stories about profitable companies that for
13
Across the whole banking system the funds on deposit should remain roughly constant
as a withdrawal from one bank usually becomes a deposit with another bank.
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Business Snapshot 15.6
Metallgesellschaft
In the early 1990s, Metallgesellschaft (MG) sold a large volume of five- to
ten-year heating oil and gasoline fixed-price supply contracts to its customers
at 6 to 8 cents above market prices. It hedged its exposure with long positions
in short-dated futures contracts that were rolled forward. As it turned out, the
price of oil fell and there were margin calls on the futures positions. Considerable short-term cash-flow pressures were placed on MG. Those at MG
who devised the hedging strategy argued that these short-term cash outflows
were offset by positive cash flows that would ultimately be realized on the
long-term fixed-price contracts. However, the company's senior management
and its bankers became concerned about the huge cash drain. As a result, the
company closed out all the hedge positions and agreed with its customers that
the fixed-price contracts would be abandoned. The outcome was a loss to
MG of $1.33 billion.
some reason "fell through the cracks" when trying to arrange venture
capital funding or bank loans. An extreme example of a liquidity funding
problem is provided by a German company, Metallgesellschaft, that
entered into profitable fixed-price oil and gas contracts with its customers
(see Business Snapshot 15.6).
Liquidity funding problems can in part be avoided by carrying out
scenario analyses and taking steps to avoid the possibility of outcomes
where short-term cash drains are difficult to fund.
SUMMARY
Since the publication of the Black-Scholes model in 1973 a huge amount
of effort has been devoted to the development of improved models for the
behavior of asset prices. It might be thought that it is just a matter of time
before the perfect model is produced. Unfortunately, this is not the case.
Models in finance are different from those in the physical sciences because
they are ultimately models of human behavior. They are always likely to
be at best approximations to the way market variables behave. Furthermore, from time to time there are regime shifts where there are fundamental changes in the behavior of market variables.
For products that trade actively, models are used primarily for communicating prices, interpolating between market prices, and hedgingWhen hedging, traders use both within-model hedging and outside-model
hedging. This means that they hedge against movements in variables that
Model Risk and Liquidity Risk
363
the model assumes to be constant (or deterministic) as well movements in
variables that are assumed to be stochastic. This type of hedging is
imperfect, but hopefully the unhedged risks are largely diversified in a
large portfolio.
For products that are highly structured or do not trade actively models
are used for pricing. In this case choosing the right model is often more of
an art than a science. It is a good practice to use several models and
assumptions about the underlying parameters in order to obtain a
realistic range for pricing and understand the accompanying model risk.
Liquidity risk is the risk that the market will not be able to absorb the
trades a financial institution wants to do at the time it wants to do them.
In normal market conditions liquidity is characterized by a bid-offer
spread. This spread widens as the size of a transaction increases.
The most serious liquidity risks arise from what are sometimes termed
liquidity black holes. These occur when all traders want to be on the same
side of the market at the same time. This may be because they are using
similar models or are subject to similar regulations, or because of a herd
mentality that sometimes develops among traders. Traders that have
long-term objectives should avoid allowing themselves to be influenced
by the short-term overreaction of markets.
FURTHER READING
Derman, E., My Life as a Quant: Reflections on Physics and Finance. New York:
Wiley, 2004.
Persaud, A. D., (ed.), Liquidity Black Holes: Understanding, Quantifying and
Managing Financial Liquidity Risk. London: Risk Books, 1999.
QUESTIONS A N D PROBLEMS (Answers at End of Book)
15.1. Give two explanations for the volatility skew observed for options on
equities.
15.2. Give two explanations for the volatility smile observed for options on a
foreign currency.
15.3. "The Black-Scholes model is nothing more than a sophisticated interpolation tool." Discuss this viewpoint.
15.4. Using Table 15.1, calculate the volatility a trader would use for an
8-month option with a strike price of 1.04.
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15.5. What is the key difference between the models of physics and the models
of finance.
15.6. How is a financial institution liable to find that it is using a model
different from its competitors for a particular type of derivatives product
15.7. What is a liquidity-adjusted VaR designed to measure?
15.8. Explain how liquidity black holes occur. How can regulation lead to
liquidity black holes?
15.9. Distinguish between within-model and outside-model hedging.
15.10. A stock price is currently $20. Tomorrow, news is expected to be
announced that will either increase the price by $5 or decrease the price
by $5. What are the problems in using Black-Scholes to value 1-month
options on the stock?
15.11. Suppose that a central bank's policy is to allow an exchange rate to
fluctuate between 0.97 and 1.03. What pattern of implied volatilities for
options on the exchange rate would you expect to see?
15.12. "For actively traded products traders can mark to market. For structured
products they mark to model." Explain this remark.
15.13. "Hedge funds can either be the solution to black holes or the cause of
black holes." Explain this remark.
ASSIGNMENT QUESTIONS
15.14. Suppose that all options traders decide to switch from Black-Scholes to
another model that makes different assumptions about the behavior of
asset prices. What effect do you think this would have on (a) the pricing
of standard options and (b) the hedging of standard options?
15.15. Using Table 15.1, calculate the volatility a trader would use for an
11-month option with a strike price of 0.98.
15.16. A futures price is currently $40. The risk-free interest rate is 5%. Some
news is expected tomorrow that will cause the volatility over the next
3 months to be either 10% or 30%. There is a 60% chance of the first
outcome and a 40% chance of the second outcome. Use the DerivaGem
software (available on the author's website) to calculate a volatility smile
for 3-month options.
Economic Capital
and RAROC
As we saw in Chapter 1, the role of capital in a bank is to protect
depositors against losses. The capital of a bank consists of common
shareholder's equity, preferred shareholder's equity, subordinated debt,
and other similar items.
In Chapter 7 we discussed the rules that the Basel Committee uses to
determine regulatory capital. These rules are the same for all banks and,
however carefully they have been chosen, it is inevitable that they will not
be exactly appropriate for any particular bank. This has led banks to
calculate economic capital (sometimes also referred to as risk capital).
Economic capital is a bank's own internal estimate of the capital it needs
for the risks it is taking. Economic capital can be regarded as a "currency" for risk-taking within a bank. A business unit can take a certain
risk only when it is allocated the appropriate economic capital for that
risk. The profitability of a business unit is measured relative to the
economic capital allocated to the unit.
In this chapter we discuss the approaches a bank uses to arrive at
estimates of economic capital for particular risk types and particular
business units and how these estimates are aggregated to produce a single
economic capital estimate for the whole bank. We also discuss riskadjusted return on capital or RAROC. This is the return earned by a
business unit on the capital assigned to it. RAROC can be used to assess
the past performance of business units. It can also be used to forecast
future performance of the units and decide on the most appropriate way
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of allocating capital in the future. It provides a basis for determining
whether some activities should be discontinued and others expanded.
16.1 DEFINITION OF ECONOMIC CAPITAL
Economic capital is defined as the amount of capital a bank needs to
absorb losses over a certain time horizon with a certain confidence level.
The time horizon is usually chosen as one year. The confidence level
depends on the bank's objectives. A common objective for a large
international bank is to maintain an AA credit rating. Corporations rated
AA have a one-year probability of default of about 0.03%. This suggests
that the confidence level should be 99.97%. For a bank wanting to
maintain a BBB credit rating the confidence level is lower. A BBB-rated
corporation has a probability of about 0.2% of defaulting in one year so
that the confidence level is 99.80%.
Capital is required to cover unexpected loss. This is defined as the
difference between the actual loss and the expected loss. The idea here
is that expected losses should be taken account of in the way a bank prices
its products so that only unexpected losses require capital. As indicated in
Figure 16.1, the economic capital for a bank that wants to maintain an
Figure 16.1
Calculation of economic capital from one-year loss distribution
for a AA-rated bank.
Ecomomic Capital and RAROC
367
AA rating is the difference between expected losses and the 99.97 percentile point on the probability distribution of losses.
Example 16.1
When lending in a certain region of the world an AA-rated bank estimates its
losses as 1% of outstanding loans per year on average. The 99.97% worst-case
loss (i.e., the loss exceeded only 0.03% of the time) is estimated as 5% of
outstanding loans. The economic capital required per $100 of loans is therefore
$4.0 (the difference between the 99.97% worst-case loss and the expected loss).
Approaches to Measurement
There are two broad approaches to measuring economic capital: the
"top-down" and "bottom-up" approaches. In the top-down approach
the volatility of the bank's assets is estimated and then used to calculate
the probability that the value of the assets will fall below the value of the
liabilities by the end of the time horizon. A theoretical framework that
can be used for the top-down approach is Merton's model, which was
discussed in Section 11.6.
The approach most often used is the bottom-up approach, where loss
distributions are estimated for different types of risk and different business units and then aggregated. The first step in the aggregation can be to
calculate probability distributions for total losses by risk type or total
losses by business unit. A final aggregation gives a probability distribution of total losses for the whole financial institution.
The various risks facing a bank are summarized in Figure 16.2. As we
saw in Chapter 14, regulators have chosen to define operational risk as
"the risk of loss resulting from inadequate or failed internal processes,
Figure 16.2 Categorization of risks faced by a bank in the Basel
II regulatory environment.
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Chapter 16
people, and systems or from external events." Operational risk includes
model risk and legal risk, but not risks arising from strategic decisions or
business reputation. We will refer to the latter risks collectively as business
risk. Regulatory capital is not required for business risk under Basel II,
but some banks do assess economic capital for business risk.
16.2 COMPONENTS OF ECONOMIC CAPITAL
In earlier chapters we covered approaches used to calculate loss distributions for different types of risks. Here we review some of the key points.
Market Risk Economic Capital
In Chapters 9 and 10 we discussed the historical simulation and modelbuilding approaches for estimating the probability distribution of the loss
or gain from market risk. As explained, this distribution is usually calculated in the first instance with a one-day time horizon. Regulatory capital
for market risk is calculated as a multiple (at least 3.0) of the ten-day 99%
VaR and bank supervisors have indicated that they are comfortable
calculating the ten-day 99% VaR as
times the one-day 99% VaR.
When calculating economic capital, we want to use the same time
horizon and confidence level for all risks. The time horizon is usually
one year and, as explained, the confidence level is often chosen as 99.97%
for an AA-rated bank. The simplest assumptions are (a) that the probability distribution of gains and losses for each day during the next year
will be the same as that estimated for the first day and (b) that the
distributions are independent. We can then use the central limit theorem
to argue that the one-year loss/gain distribution is normal. Assuming 252
business days in the year, the standard deviation of the one-year loss/gain
equals the standard deviation of the daily loss/gain multiplied by
The mean loss/gain is much more difficult to estimate than the standard
deviation. A reasonable, if somewhat conservative, assumption is that the
mean loss/gain is zero. The 99.97% worst-case loss is then 3.43 times the
standard deviation of the one-year loss/gain. The 99.8% worst-case loss is
2.88 times the standard deviation of the one-year loss/gain.
Example 16.2
Suppose that the one-day standard deviation of market risk losses/gains for a
bank is $5 million. The one-year 99.8% worst-case loss is 2.88 x
x5=
228.6, or $228.6 million.
Economic
Capital
and RAROC
369
Note that we are not assuming that the daily losses/gains are normal.
All we are assuming is that they are independent and identically
distributed. The central limit theorem of statistics tells us that the
sum of many independent identically distributed variables is approximately normal. If losses on successive days are correlated, we can
assume first-order autocorrelation, estimate the correlation parameter
from historical data, and use the results in Section 8.4. When the
autocorrelation is not too high, it is still reasonable to assume that
the one-year loss distribution is normal. If a more complicated model
for the relationship between losses on successive days is considered
appropriate, then the one-year loss distribution can be calculated using
Monte Carlo simulation.
Credit Risk Economic Capital
Although Basel II gives banks that use the internal ratings based approach
for regulatory capital a great deal of freedom, it does not allow them to
choose their own credit correlation model and correlation parameters.
When calculating economic capital, banks are free to make the assumptions they consider most appropriate for their situation. As explained in
Section 12.6, CreditMetrics is often used to calculate the specific risk
capital charge for credit risk in the trading book. It is also sometimes used
when economic capital is calculated for the banking book. A bank's own
internal rating system can be used instead of that of Moody's or S&P when
this method is used.
Another approach that is sometimes used is Credit Risk Plus, which is
described in Section 12.5. This approach borrows a number of ideas from
actuarial science to calculate a probability distribution for losses from
defaults. Whereas CreditMetrics calculates the loss from downgrades and
defaults, Credit Risk Plus calculates losses from defaults only.
In calculating credit risk economic capital, a bank can choose to adopt
a conditional or unconditional model. In a conditional (cycle-specific)
model, the expected and unexpected losses take account of current
economic conditions. In an unconditional (cycle-neutral) model, they
are calculated by assuming economic conditions that are in some sense
an average of those experienced through the cycle. Rating agencies aim
to produce ratings that are unconditional. Moreover, when regulatory
capital is calculated using the internal ratings based approach, the PD
and LGD estimates should be unconditional. Obviously it is important
to be consistent when economic capital is calculated. If expected losses
370
Chapter 16
are conditional, unexpected losses should also be conditional. If expected losses are unconditional, the same should be true of unexpected
losses.
A particularly challenging task is to take counterparty risk on derivatives into account when credit risk loss distributions are calculated. In
practice, banks often use approximations to the approach outlined in
Section 12.1. For example, they might develop look-up tables for expected
exposure during the life of an instrument and assume that exposure
remains constant at this level. When a bank has several different exposures
with the same counterparty and there are netting agreements, algorithms
for calculating expected exposure can be developed. Other features of
derivative contracts such as Collateralization and downgrade triggers can
be incorporated.
Operational Risk Economic Capital
Banks are given a great deal of freedom in the assessment of regulatory
capital for operational risk under the advanced measurement approach. It
is therefore likely that most banks using this approach will calculate
operational risk economic capital and operational risk regulatory capital
in the same way. As noted in Chapter 14, methods for calculating
operational risk capital are still evolving. Some approaches are statistical
and others are more subjective.
Business Risk Economic Capital
As mentioned earlier, business risk includes strategic risk (relating to a
bank's decision to enter new markets and develop new products) and
reputational risk. Business risk is even more difficult to quantify than
operational risk and estimates are likely to be largely subjective. It is
important that senior risk managers within a financial institution have a
good understanding of the portfolio of business risks being taken. This
should enable them to assess the capital required for the risks and, more
importantly, the marginal impact on total risk of new strategic initiatives
that are being contemplated.
16.3 SHAPES OF THE LOSS DISTRIBUTIONS
The loss probability distributions for market, credit, and operational risk
are very different. Rosenberg and Schuermann used data from a variety of
Economic Capital and RAROC
Figure 16.3
371
Loss density distribution for market risk.
different sources to estimate typical shapes for these distributions.1 These
are shown in Figures 16.3, 16.4, and 16.5. The market risk loss distribution
(see Figure 16.3) is symmetrical, but not perfectly normally distributed. A
t-distribution with 11 degrees of freedom provides a good fit. The credit
risk loss distribution in Figure 16.4 is quite skewed, as one would expect.
The operational risk distribution in Figure 16.5 has a quite extreme shape.
Most of the time losses are modest, but occasionally they are very large.
We can characterize a distribution by its second, third, and fourth
moments. Loosely speaking, the second moment measures standard
deviation (or variance), the third Skewness, and the fourth kurtosis
Figure 16.4
Loss density distribution for credit risk.
1
See J. V. Rosenberg and T. Schuermann, "A General Approach to Integrated Risk
Management with Skewed, Fat-Tailed Risks," Federal Reserve Bank of New York, Staff
Report No. 185, May 2004.
372
Chapter 16
Figure 16.5
Loss density distribution for
operational risk.
(i.e., the heaviness of tails). Table 16.1 summarizes the properties of
typical loss distributions for market, credit, and operational risk.
16.4 RELATIVE IMPORTANCE OF RISKS
The relative importance of different types of risks depends on the business
mix. For a bank whose prime business is taking deposits and making
loans, credit risk is of paramount importance. For an investment bank,
credit risk and market risk are both important. For an asset manager, the
greatest risk is operational risk. If rules on the ways funds are to be
invested are not followed, there are liable to be expensive investor law
suits. Business Snapshot 16.1 gives one example of this. Another highprofile example is provided by the Unilever's pension plan. Mercury Asset
Management, owned by Merrill Lynch, pledged not to underperform a
benchmark index by more than 3%. Between January 1997 and March
1998 it underperformed the index by 10.5%. Unilever sued Merrill Lynch
for $185 million and the matter was settled out of court.
Table 16.1
Market risk
Credit risk
Operational risk
Characteristics of loss distributions for different risk types.
Second moment
(standard deviation)
Third moment
(skewness)
Fourth moment
(kurtosis)
High
Moderate
Low
Zero
Moderate
High
Low
Moderate
High
Economic
Capital
373
and RAROC
Business Snapshot 16.1
The EGT Fund
In 1996 Peter Young was fund manager at Deutsche Morgan Grenfell, a
subsidiary of Deutsche Bank. He was responsible for managing a fund called
the European Growth Trust (EGT). It had grown very large and Young had
responsibilities for managing over one billion pounds of investors' money.
Certain rules applied to EGT. One of these was that no more than 10% of
the fund could be invested in unlisted securities. Peter Young violated this rul
in a way, it can be argued, that benefited him personally. When the facts were
uncovered, he was fired and Deutsche bank had to compensate investors. The
total cost to Deutsche Bank was over 200 million pounds.
Interactions between Risks
There are interactions between the different types of risk. For example,
when a derivative such as a swap is traded, there are interactions between
credit and market risk. If the counterparty defaults, credit risk exists only
if market variables have moved so that the value of the derivative to the
financial institution is positive. Another interaction is that the probability
of default by a counterparty may depend on the value of a financial
institution's contract (or contracts) with the counterparty. 2 If the counterparty has entered into the contract for hedging purposes, this is not likely
to be the case. However, if the contract has been entered into for
speculative purposes and the contract is large in relation to the size of
the counterparty, there is likely to be some dependence.
As the Long-Term Capital Management saga clearly shows there can
be interactions between liquidity risks and market risks (see Section 15.9).
There are also interactions between operational risks and market risks. It
is unlikely that we would know about the activities of Nick Leeson at
Barings Bank if he had guessed right about the Nikkei index (see Business
Snapshot 2.3). It is also unlikely that we would hear about a violation of
the rules for a fund (such as the one in Business Snapshot 16.1) if it had
led to a gain rather than a loss.
16.5 AGGREGATING ECONOMIC CAPITAL
Typically a financial institution calculates market, credit, operational, and
(possibly) business risk loss distributions for a number of different business
2
When calculating the expected cost of counterparty default risk in Section 12.1, we
(assumed no dependence.
374
Chapter 16
units. It is then faced with the problem of aggregating the loss distributions
to calculate a total economic capital for the whole enterprise.
The simplest approach is to assume that the total economic capital for a
set of n different risks is the sum of the economic capital amounts for each
risk considered separately so that
where
is the total economic capital for the financial institution
facing n different risks and
is the economic capital for the ith risk
considered on its own. This is what the Basel Committee does for
regulatory capital. The total regulatory capital a bank is required to keep
under Basel II is the sum of the regulatory capital amounts for credit,
market, and operational risks.
Equation (16.1) is clearly a very conservative assumption. It assumes
perfect correlation. In the context of economic capital calculations where
the confidence level is 99.97%, it would mean that, if a financial institution
experiences the 99.97% worst-case loss for market risk, it also experiences
the 99.97% worst-case loss for credit risk and operational risk. Rosenberg
and Schuermann estimate the correlation between market risk and credit
risk to be approximately 50 % and the correlation between each of these
risks and operational risk to be approximately 20%. They estimate that
equation (16.1), when used as a way of aggregating market, credit, and
operational risk, overstates the total capital required by about 40%.
Assuming Normal Distributions
A simple assumption when aggregating loss distributions is that they are
normally distributed. The standard deviation of the total loss from
n sources of risk is then
where
is the standard deviation of the loss from the ith source of risk
and
is the correlation between risk i and risk j. The capital requirement can be calculated from this. For example, the excess of the 99.97%
worst-case loss over the expected loss is 3.44 times the number calculated
in equation (16.2).
This approach tends to underestimate the capital requirement because
it takes no account of the Skewness and kurtosis of the loss distributions.
Economic Capital and RAROC
375
Rosenberg and Schuermann estimate that, when the approach is applied
to aggregating market, credit, and operational risk, the total capital is
underestimated by about 40%.
Using Copulas
A more sophisticated approach to aggregating loss distributions is by using
copulas. Copulas were discussed in Chapter 6. Each loss distribution is
mapped on a percentile-to-percentile basis to a standard well-behaved
distribution. A correlation structure between the standard distributions
is defined and this indirectly defines a correlation structure between the
original distributions.
Many different copulas can be defined. In the Gaussian copula the
standard distributions are assumed to be multivariate normal. An alternative is to assume that they are multivariate t. This leads to the joint
probability of extreme values of two variables being higher than in the
Gaussian copula. This is discussed further in Section 6.4.
The Hybrid Approach
A simple approach that seems to work well is known as the hybrid
approach. This involves calculating the economic capital for a portfolio
of risks from the economic capital for the individual risks using
When the distributions are normal, this approach is exactly correct. When
they are nonnormal, the hybrid approach gives an approximate answer—
but one that reflects any heaviness in the tails of the individual loss
distributions. Rosenberg and Schuermann find that the answers given by
the hybrid approach are reasonably close to those given by copula models.
Example 16.3
Suppose that the estimates for economic capital for market, credit, and
operational risk for two business units are as shown in Table 16.2. The
correlations between the losses are shown in Table 16.3. The correlation
between credit risk and market risk within the same business unit is 0.5,
and the correlation between operational risk and either credit or market risk
within the same business unit is 0.2. (These correspond to the estimates of
Rosenberg and Schuermann mentioned above.) The correlation between two
different risk types in two different business units is zero. The correlation
between market risks across business units is 0.4. The correlation between
376
Chapter 16
Table 16.2
Economic capital estimates for
Example 16.3.
Business unit
Market risk
Credit risk
Operational risk
1
2
30
70
30
40
80
90
credit risk across business units is 0.6. The correlation between operational
risk across business units is zero.
We can aggregate the economic capital in a number of ways. The total
market risk economic capital is
The total credit risk economic capital is
The total operational risk economic capital is
The total economic capital for Business Unit 1 is
The total economic capital for Business Unit 2 is
Table 16.3 Correlations between losses in Example 16.3.
MR, CR, and OR refer to market risk, credit risk, and
operational risk; 1 and 2 refer to business units.
MR-1
CR-1
OR-1
MR-2
CR-2
OR-2
MR-1
CR-1
OR-1
MR-2
CR-2
OR-2
1.0
0.5
0.2
0.4
0.0
0.0
0.5
1.0
0.2
0.0
0.6
0.0
0.2
0.2
1.0
0.0
0.0
0.0
0.4
0.0
0.0
1.0
0.5
0.2
0.0
0.6
0.0
0.5
1.0
0.2
0.0
0.0
0.0
0.2
0.2
1.0
Economic Capital and RAROC
377
The total enterprise-wide economic capital is the square root of
302 + 402 + 702 + 802 + 302 + 902 + 2 x 0.4 x 30 x 40 + 2 x 0.5 x 30 x 70
+ 2 x 0.2 x 30 x 30 + 2 x 0.5 x 40 x 80 + 2 x 0.2 x 40 x 90
+ 2 x 0.6 x 70 x 80 + 2 x 0.2 x 70 x 30 + 2 x 0.2 x 80 x 90
or 203.224.
There are significant diversification benefits. The sum of the economic capital
estimates for market, credit, and operational risk is 58.8 + 134.2 + 94.9 = 287.9
and the sum of the economic capital estimates for two business units is
100 + 153.7 = 253.7. Both of these are greater than the total economic capital
estimate of 203.2.
16.6 ALLOCATION OF THE DIVERSIFICATION
BENEFIT
Suppose that the sum of the economic capital for each business unit,
is $2 billion and the total economic capital for the whole bank,
after taking less-than-perfect correlations into account, is $1.3 billion
(= 65% of the sum of the E's). The $0.7 billion is a diversification gain
to the bank. How should it be allocated to the business units?
A simple approach is to reduce the economic capital of each business
unit by 35%. However, this is probably not the best approach. Suppose
there are 50 business units and that two particular business units both
have an economic capital of $100 million. Suppose that when the first
business unit is excluded from the calculations the bank's economic
capital reduces by $60 million and that when the second business unit
is excluded from the calculation the bank's economic capital reduces by
$10 million. Arguably the first business unit should have more economic
capital than the second business unit because its incremental impact on
the bank's total economic capital is greater.
The issues here are analogous to the issues we discussed in Section 8.5
concerned with allocating VaR. The theoretically best allocation scheme
is to allocate an amount
to the ith business unit, where E is the total economic capital and is the
investment in the ith business unit. As we pointed out in Section 8.5, a
result known as Euler's theorem ensures that the total of the allocated
capital is E.
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Chapter 16
Define
as the increase in the total economic capital when we
increase
by
A discrete approximation for the amount allocated
to business unit i is
where
Example 16.4
Consider again Example 16.3. The total economic capital is 203.2. The
economic capital calculated for Business Unit 1 is 100 and that calculated
for Business Unit 2 is 153.7.
A naive procedure would allocate 100/253.7 of the total economic capital to
Business Unit 1 and 153.7/253.7 of the economic capital to Business Unit 2.
This would result in 80.1 for Business Unit 1 and 123.1 for Business Unit 2.
The incremental effect of Business Unit 1 on the total economic capital is
203.2 - 153.7 = 49.5. Similarly, the incremental effect of Business Unit 2 is
203.2 - 100 = 103.2. The two incremental capitals do not add up to the total
capital (as is usually the case). We could use them as a basis for allocating the
total capital. We would then allocate 49.5/(49.5+ 103.2) of the capital to
Business Unit 1 and 103.2/(49.5 + 103.2) of it to Business Unit 2. This would
result in 65.9 for Business Unit 1 and 137.3 for Business Unit 2.
To apply equation (16.4), we could calculate the partial derivative analytically. Alternatively, we can use a numerical approximation. When we increase
the size of Business Unit 1 by 1%, its economic capital amounts for market,
credit, and operational risk in Table 16.2 increase to 30.3, 70.7, and 30.3,
respectively. The total economic capital becomes 203.906, so that
=
203.906-203.224 = 0.682.
When we increase the size of Business Unit 2 by 1%, its economic capital
amounts for market, credit, and operational risk in Table 16.2 increase to 40.4,
80.8, and 90.9, respectively. The total economic capital becomes 204.577, so
that
= 204.577 - 203.224 = 1.353.
In this case, because we are considering 1 % increases in the size of each unit,
= 0.01. From equation (16.4) the economic capital allocations to the two business units are 68.2 and 135.2. (These do not quite add up
to the total economic capital of 203.2 because we approximated the partial
derivative.)
16.7 DEUTSCHE BANK'S ECONOMIC CAPITAL
Deutsche Bank publishes the result of its economic capital calculation in
its annual financial statements. Table 16.4 summarizes the economic
capital and regulatory capital for 2004. Deutsche Bank calculated a
Economic Capital and RAROC
379
Table 16.4 Deutsche Bank's economic capital and regulatory
capital (millions of euros).
Credit risk
Market risk
Diversification benefit across credit and market risk
Operational risk
Business risk
Total economic capital
Total risk-weighted assets
Tier 1 capital held (% of risk-weighted assets)
Tier 2 capital held (% of risk-weighted assets)
Total capital held (% of risk-weighted assets)
5,971
5,476
(870)
2,243
381
13,201
216,787
8.6%
4.6%
13.2%
diversification benefit for credit and market risk, but not for other risktype combinations. The total economic capital is about 13.2 billion euros.
This is considerably less than the total regulatory capital which is 8% of
216.8, or about 17.3 billion euros. The actual capital held is about
18.6 billion euros of Tier 1 capital and 10.0 billion euros of Tier 2 capital.
It would appear that Deutsche Bank is very well capitalized relative to the
risks it is taking.
16.8 RAROC
Risk-adjusted performance measurement (RAPM) has become an important part of how business units are assessed. There are many different
approaches, but all have one thing in common. They compare return with
capital employed in a way that incorporates an adjustment for risk.
The most common approach is to compare expected return with
economic capital. This is usually referred to as RAROC (risk-adjusted
return on capital). The formula is
The numerator may be calculated on a pre-tax or post-tax basis. Very
often, a risk-free rate of return on the economic capital is calculated and
added to the numerator.
380
Chapter 16
Example 16.5
When lending in a certain region of the world, an AA-rated bank estimates its
losses as 1 % of outstanding loans per year on average. The 99.97% worst-case
loss (i.e., the loss exceeded only 0.03% of the time) is 5% of outstanding loans.
As shown in Example 16.1, the economic capital required per $100 of loans is
$4, which is the difference between the 99.97% worst-case loss and the expected
loss. (This ignores diversification benefits that would in practice be allocated to
the lending.) The spread between the cost of funds and the interest charged is
2.5%. Subtracting from this the expected loan loss of 1%, the expected profit
per $100 of loans is $1.50. Assume that the lending department's administrative
costs total 0.7% of the amount loan, the expected profit is reduced to $0.80 per
$100 in the loan portfolio. RAROC is therefore
An alternative calculation would add the interest on the economic capital to the
numerator. Suppose the risk-free interest rate is 2%. Then 0.02 x 4 = 0.08 is
added to the numerator, so that RAROC becomes
As pointed out by Matten, it is more accurate to refer to the approach in
equation (16.5) as RORAC (return on risk-adjusted capital) rather than
RAROC. 3 In theory, RAROC should involve adjusting the return (i.e.,
the numerator) for risk. In equation (16.5) it is the capital (i.e., the
denominator) that is adjusted for risk.
There are two ways in which RAROC is used. One is as a tool to
compare the past performance of different business units, decide on endof-the-year bonuses, etc. The other is as a tool to decide whether a
particular business unit should be expanded or contracted. The latter
involves predicting an average RAROC for the unit and comparing it
with the bank's threshold return on capital.
When RAROC is used for the second purpose, it should be noted that
it could be low simply because the business unit had a bad year. Perhaps
credit losses were much larger than average or there was an unexpectedly
large operational risk loss. This is not necessarily an indication that the
business unit should be shut down. When RAROC is used as a forwardlooking measure, the calculation should reflect average losses. The aim is
to assess the long-term viability of the business unit, whether it should be
expanded or scaled back, and so on.
3
See C. Matten, Managing Bank Capital: Capital Allocation and Performance
Measurement, 2nd edn., Chichester, UK: Wiley, 2000.
Economic Capital and RAROC
381
SUMMARY
Economic capital is the capital that a bank or other financial institution
deems necessary for the risks it is bearing. When calculating economic
capital, a financial institution is free to adopt any approach it likes. It
does not have to use the one proposed by regulators. Typically, it
estimates economic capital for credit risk, market risk, operational risk,
and (possibly) business risk for its business units and then aggregates
these estimates to produce an estimate of the economic capital for the
whole enterprise. The one-year loss distributions for market risk, credit
risk and operational risk are quite different. The loss distribution for
market risk is symmetrical. For credit risk, it is skewed and for operational risk it is highly skewed with very heavy tails.
The total economic capital for a financial institution is allocated to
business units so that a return on capital can be calculated. There are a
number of allocation schemes. The best are those that reflect the incremental impact of the business unit on the total economic capital. The
amount of capital allocated to a business unit is generally less than the
capital estimated for the business unit as a stand-alone entity because of
diversification benefits.
FURTHER READING
Dev, A., Economic Capital: A Practitioner's Guide. London: Risk Books, 2004.
Matten, C, Managing Bank Capital: Capital Allocation and Performance
Measurement, 2nd edn. Chichester, UK: Wiley, 2000.
Rosenberg, J. V., and T. Schuermann, "A General Approach to Integrated Risk
Management with Skewed, Fat-Tailed Risks," Federal Reserve Bank of New
York, Staff Report No. 185, May 2004.
QUESTIONS A N D PROBLEMS (Answers at End of Book)
16.1. What is the difference between economic capital and regulatory capital?
16.2. Why do AA-rated banks use a confidence level of 99.97% when calculating economic capital for a one-year time horizon?
16.3. What is included in business risk?
16.4. In what respects are the models used to calculate economic capital for
market risk, credit risk, and operational risk likely to be different from
those used to calculate regulatory capital?
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Chapter 16
16.5. Suppose the credit loss in a year has a lognormal distribution. The logarithm of the loss is normal with mean 0.5 and standard deviation 4. What
is the economic capital requirement if a confidence level of 99.97% is used?
16.6. Suppose that the economic capital estimates for two business units are as
follows:
Business unit
1
2
Market risk
Credit risk
Operational risk
20
40
70
40
30
10
The correlations are as in Table 16.3. Calculate the total economic capital
for each business unit and the two business units together.
16.7. In Problem 16.6, what is the incremental effect of each business unit on
the total economic capital? Use this to allocate economic capital to
business units. What is the impact on the economic capital of each
business unit increasing by 0.5%? Show that your results are consistent
with Euler's theorem.
16.8. A bank is considering expanding its asset management operations. The
main risk is operational risk. It estimates that the expected operational
risk loss from the new venture in one year is $2 million and the 99.97%
worst-case loss (arising from a large investor law suit) is $40 million. The
expected fees it will receive from investors for the funds under administration are $12 million per year and administrative costs are expected to
be $5 million per year. Estimate the before-tax RAROC?
16.9. RAROC can be used in two different ways. What are they?
ASSIGNMENT QUESTIONS
16.10. Suppose that daily gains and losses are normally distributed with a
standard deviation of
(a) Estimate the minimum regulatory capital
the bank is required to hold for market risk. (Assume a multiplicative
factor of 3.0.) (b) Estimate the economic capital for market risk using a
one-year time horizon and a 99.97% confidence limit. (c) Why is the ratio
of your answer in (a) to your answer in (b) not a good indication of the
ratio of regulatory market risk capital to economic market risk capital in
practice?
16.11. Suppose that a bank's sole business is to lend in two regions of the world.
The lending in each region has the same characteristics as in Example 16.5
of Section 16.8. Lending to Region A is three times as great as lending to
Economic Capital and RAROC
383
Region B. The correlation between loan losses in the two regions is 0.4.
Estimate the total RAROC.
16.12. Suppose that the economic capital estimates for two business units are as
follows:
Business unit
Market risk
Credit risk
Operational risk
1
2
10
30
50
50
30
10
The correlation between market risk and credit risk in the same business
unit is 0.3. The correlation between credit risk in one business unit and
credit risk in another is 0.7. The correlation between market risk in one
business unit and market risk in the other is 0.2. All other correlations are
zero. Calculate the total economic capital. How much should be allocated
to each business unit?
Weather, Energy,
and Insurance
Derivatives
Products to manage equity risks, interest rate risks, and foreign currency
risks are well established and were covered in Chapter 2. This chapter
examines the products that have been developed to manage risk in less
traditional markets. Specifically, it considers weather risk, energy price
risk, and insurance risks. The markets that we will talk about are in some
cases in the early stages of their development. As they evolve, we may well
see significant changes in both the products that are offered and the ways
in which they are used.
17.1 WEATHER DERIVATIVES
Many companies are in the position where their performance is liable to
be adversely affected by the weather.1 It makes sense for these companies
to consider hedging their weather risk in much the same way as they
hedge foreign exchange or interest rate risks.
The first over-the-counter weather derivatives were introduced in 1997.
To understand how they work, we explain two variables:
HDD: Heating degree days
CDD: Cooling degree days
1
The US Department of Energy has estimated that one-seventh of the US economy is
subject to weather risk.
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A day's HDD is defined as
HDD = max(0, 65 - A)
and a day's CDD is defined as
CDD = max(0, A - 65)
where A is the average of the highest and lowest temperature during the
day at a specified weather station, measured in degrees Fahrenheit. For
example, if the maximum temperature during a day (midnight to midnight) is 68° Fahrenheit and the minimum temperature is 44° Fahrenheit,
A = 56. The daily HDD is then 9 and the daily CDD is 0.
A typical over-the-counter product is a forward or option contract
providing a payoff dependent on the cumulative HDD or CDD during
a month (i.e., the total of the HDDs or CDDs for every day in the
month). For example, a dealer could in January 2007 sell a client a call
option on the cumulative HDD during February 2008 at the Chicago
O'Hare Airport weather station with a strike price of 700 and a payment
rate of $10,000 per degree day. If the actual cumulative HDD is 820, the
payoff is $1.2 million. Often contracts include a payment cap. If the cap in
our example is $1.5 million, the client's position is equivalent to a long
call option on cumulative HDD with a strike price of 700 and a short call
option with a strike price of 850.
A day's HDD is a measure of the volume of energy required for heating
during the day. A day's CDD is a measure of the volume of energy
required for cooling during the day. Most weather derivative contracts are
entered into by energy producers and energy consumers. But retailers,
supermarket chains, food and drink manufacturers, health service companies, agriculture companies, and companies in the leisure industry are
also potential users of weather derivatives. The Weather Risk Management Association (www.wrma.org) has been formed to serve the interests
of the weather risk management industry.
In September 1999, the Chicago Mercantile Exchange began trading
weather futures and European options on weather futures. The contracts
are on the cumulative HDD and CDD for a month observed at a weather
station.2 The contracts are settled in cash just after the end of the month
once the HDD and CDD are known. One futures contract is on $100 times
2
The CME has introduced contracts for ten different weather stations (Atlanta,
Chicago, Cincinnati, Dallas, Des Moines, Las Vegas, New York, Philadelphia, Portland,
and Tucson).
Weather, Energy, and Insurance Derivatives
387
the cumulative HDD or CDD. The HDD and CDD are calculated by a
company, Earth Satellite Corporation, using automated data collection
equipment.
17.2 ENERGY DERIVATIVES
Energy companies are among the most active and sophisticated users of
derivatives. Many energy products trade in both the over-the-counter
market and on exchanges. In this section we will examine the trading in
crude oil, natural gas, and electricity derivatives.
Crude Oil
Crude oil is one of the most important commodities in the world with
global demand amounting to about 80 million barrels daily. Ten-year
fixed-price supply contracts have been commonplace in the over-thecounter market for many years. These are swaps where oil at a fixed
price is exchanged for oil at a floating price.
In the 1970s the price of oil was highly volatile. The 1973 war in the
Middle East led to a tripling of oil prices. The fall of the Shah of Iran in
the late 1970s again increased prices. These events led oil producers and
users to a realization that they needed more sophisticated tools for
managing oil price risk. In the 1980s both the over-the-counter market
and the exchange-traded market developed products to meet this need.
In the over-the-counter market, virtually any derivative that is available
on common stocks or stock indices is now available with oil as the
underlying asset. Swaps, forward contracts, and options are popular.
Contracts sometimes require settlement in cash and sometimes require
settlement by physical delivery (i.e., by delivery of the oil).
Exchange-traded contracts are also popular. The New York Mercantile
Exchange (NYMEX) and the International Petroleum Exchange (IPE)
trade a number of oil futures and futures options contracts. Some of the
futures contracts are settled in cash; others are settled by physical
delivery. For example, the Brent crude oil futures traded on the IPE
has cash settlement based on the Brent index price; the light sweet crude
oil futures traded on NYMEX requires physical delivery. In both cases
the amount of oil underlying one contract is 1,000 barrels. NYMEX also
trades popular contracts on two refined products: heating oil and gasoline. In both cases one contract is for the delivery of 42,000 gallons.
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Natural Gas
The natural gas industry throughout the world has been going through a
period of deregulation and the elimination of government monopolies
The supplier of natural gas is now not necessarily the same company as
the producer of the gas. Suppliers are faced with the problem of meeting
daily demand.
A typical over-the-counter contract is for the delivery of a specified
amount of natural gas at a roughly uniform rate over a one-month
period. Forward contracts, options, and swaps are available in the
over-the-counter market. The seller of gas is usually responsible for
moving the gas through pipelines to the specified location.
NYMEX trades a contract for the delivery of 10,000 million British
thermal units of natural gas. The contract, if not closed out, requires
physical delivery to be made during the delivery month at a roughly
uniform rate to a particular hub in Louisiana. The IPE trades a similar
contract in London.
Electricity
Electricity is an unusual commodity because it cannot easily be stored.3
The maximum supply of electricity in a region at any moment is
determined by the maximum capacity of all the electricity-producing
plants in the region. In the United States there are 140 regions known
as control areas. Demand and supply are first matched within a control
area, and any excess power is sold to other control areas. It is this excess
power that constitutes the wholesale market for electricity. The ability of
one control area to sell power to another control area depends on the
transmission capacity of the lines between the two areas. Transmission
from one area to another involves a transmission cost, charged by the
owner of the line, and there are generally some energy transmission losses.
A major use of electricity is for air-conditioning systems. As a result the
demand for electricity, and therefore its price, is much greater in the
summer months than in the winter months. The nonstorability of electricity causes occasional very large movements in the spot price. Heat
waves have been known to increase the spot price by as much as 1000%
for short periods of time.
Like natural gas, electricity has been going through a period of
3
Electricity producers with spare capacity sometimes use it to pump water to the top of
their hydroelectric plants so that it can be used to produce electricity at a later time. This
is the closest they can get to storing this commodity.
Weather, Energy, and Insurance Derivatives
389
deregulation and the elimination of government monopolies. This has
been accompanied by the development of an electricity derivatives market. NYMEX now trades a futures contract on the price of electricity, and
there is an active over-the-counter market in forward contracts, options,
and swaps. A typical contract (exchange-traded or over-the-counter)
allows one side to receive a specified number of megawatt hours for a
specified price at a specified location during a particular month. In a
5 x 8 contract, power is received for five days a week (Monday to Friday)
during the off-peak period (11 p.m. to 7 a.m.) for the specified month. In
a 5 x 16 contract, power is received five days a week during the on-peak
period (7 a.m. to 11 p.m.) for the specified month. In a 7 x 24 contract, it
is received around the clock every day during the month. Option contracts have either daily exercise or monthly exercise. In the case of daily
exercise, the option holder can choose on each day of the month (by
giving one day's notice) to receive the specified amount of power at the
specified strike price. When there is monthly exercise a single decision on
whether to receive power for the whole month at the specified strike price
is made at the beginning of the month.
An interesting contract in electricity and natural gas markets is what is
known as a swing option or take-and-pay option. In this contract a
minimum and maximum for the amount of power that must be purchased
at a certain price by the option holder is specified for each day during a
month and for the month in total. The option holder can change (or
swing) the rate at which the power is purchased during the month, but
usually there is a limit on the total number of changes that can be made.
How an Energy Producer Can Hedge Risks
There are two components to the risks facing an energy producer. One is
the price risk; the other is the volume risk. Although prices do adjust to
reflect volumes, there is a less-than-perfect relationship between the two,
and energy producers have to take both into account when developing a
hedging strategy. The price risk can be hedged using the energy derivative
contracts discussed in this section. The volume risks can be hedged using
the weather derivatives discussed in the previous section.
Define:
Y: Profit for a month
P: Average energy prices for the month
T: Relevant temperature variable (HDD or CDD) for the month
An energy producer can use historical data to obtain a best-fit linear
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regression relationship of the form
where is the error term. The energy producer can then hedge risks for
the month by taking a position of —b in energy forwards or futures and a
position of — c in weather forwards or futures. The relationship can also
be used to analyze the effectiveness of alternative option strategies.
17.3 INSURANCE DERIVATIVES
When derivative contracts are used for hedging purposes, they have many
of the same characteristics as insurance contracts. Both types of contracts
are designed to provide protection against adverse events. It is therefore
not surprising that many insurance companies have subsidiaries that
trade derivatives and that many of the activities of insurance companies
are becoming very similar to those of investment banks.
Traditionally the insurance industry has hedged its exposure to catastrophic (CAT) risks such as hurricanes and earthquakes using a practice
known as reinsurance. Reinsurance contracts can take a number of forms.
Suppose that an insurance company has an exposure of $100 million to
earthquakes in California and wants to limit this to $30 million. One
alternative is to enter into annual reinsurance contracts that cover on a
pro rata basis 70% of its exposure. If California earthquake claims in a
particular year total $50 million, the costs to the company would then be
only 0.3 x $50 or $15 million. Another more popular alternative, involving
lower reinsurance premiums, is to buy a series of reinsurance contracts
covering what are known as excess cost layers. The first layer might provide
indemnification for losses between $30 million and $40 million; the next
layer might cover losses between $40 million and $50 million; and so on.
Each reinsurance contract is known as an excess-of-loss reinsurance contract. The insurance company is long a call option with a strike price equal
to the lower end of the layer and short a call option with a strike price equal
to the upper end of the layer.
The principal providers of CAT reinsurance have traditionally been
reinsurance companies and Lloyds syndicates (unlimited liability syndicates of wealthy individuals). In recent years the industry has come to the
conclusion that its reinsurance needs have outstripped what can be provided from these traditional sources. It has searched for new ways in which
capital markets can provide reinsurance. One of the events that caused the
Weather, Energy, and Insurance Derivatives
391
industry to rethink its practices was Hurricane Andrew in 1992. This
caused about $15 billion of insurance costs in Florida, which exceeded
the total of relevant insurance premiums received there during the previous
seven years. If the hurricane had hit Miami, it is estimated that insured
losses would have exceeded $40 billion. Hurricane Andrew and other
catastrophes have led to increases in insurance/reinsurance premiums.
Exchange-traded insurance futures contracts have been developed by
the CBOT, but have not been highly successful. The over-the-counter
market has come up with a number of products that are alternatives to
traditional reinsurance. The most popular is a CAT bond. This is a bond
issued by a subsidiary of an insurance company that pays a higher-thannormal interest rate. In exchange for the extra interest, the holder of the
bond agrees to provide an excess-of-cost reinsurance contract. Depending
on the terms of the CAT bond, the interest or principal (or both) can be
used to meet claims. In the example considered above where an insurance
company wants protection for California earthquake losses between
$30 million and $40 million, the insurance company could issue CAT
bonds with a total principal of $10 million. In the event that the insurance
company's California earthquake losses exceed $30 million, bondholders
will lose some or all of their principal. As an alternative, the insurance
company could cover this excess cost layer by making a much bigger
bond issue where only the bondholders' interest is at risk.
CAT bonds typically give a high probability of an above-normal rate of
interest and a low-probability of a high loss. Why would investors be
interested in such instruments? The answer is that there are no statistically
significant correlations between CAT risks and market returns. 4 CAT
bonds are therefore an attractive addition to an investor's portfolio. They
have no systematic risk, so that their total risk can be completely
diversified away in a large portfolio. If a CAT bond's expected return is
greater than the risk-free interest rate (and typically it is), it has the
potential to improve risk-return trade-offs.
SUMMARY
When there are risks to be managed, markets have been very innovative in
developing products to meet the needs of market participants.
4
See R. H. Litzenberger, D. R. Beaglehole, and C. E. Reynolds, "Assessing Catastrophe
Reinsurance-Linked Securities as a New Asset Class," Journal of Portfolio Management,
Winter 1996, 76-86.
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In the weather derivatives market, two measures, HDD and CDD, have
been developed to describe the temperature during a month. These are
used to define the payoffs on both exchange-traded and over-the-counter
derivatives. As the weather derivatives market develops, we will see contracts on rainfall, snow, and similar variables become more commonplace.
In energy markets, oil derivatives have been important for some time
and play a key role in helping oil producers and oil consumers manage
their price risk. Natural gas and electricity derivatives are relatively new.
They became important for risk management when these markets were
deregulated and government monopolies discontinued.
Insurance derivatives are beginning to become an alternative to traditional reinsurance as a way for insurance companies to manage the risks of
a catastrophic event such as a hurricane or an earthquake. No doubt we
will see other sorts of insurance (e.g., life insurance and automobile
insurance) being securitized in a similar way as this market develops.
FURTHER READING
On Weather Derivatives
Arditti, F., L. Cai, M. Cao, and R. McDonald, "Whether to Hedge," Risk,
Supplement on Weather Risk, 1999: 9-12.
Cao, M., and J. Wei, "Weather Derivatives Valuation and the Market Price of
Weather Risk," Journal of Futures Markets, 24, 11 (November 2004),
1065-1089.
Hunter, R., "Managing Mother Nature," Derivatives Strategy, February 1999.
On Energy Derivatives
Clewlow, L., and C. Strickland, Energy Derivatives: Pricing and Risk Management. Lacima Group, 2000.
Eydeland, A., and H. Geman, "Pricing Power Derivatives." Risk, October 1998:
71-73.
Joskow, P., "Electricity Sectors in Transition," The Energy Journal, 19 (1998):
25-52.
Kendall, R., "Crude Oil: Price Shocking," Risk, Supplement on Commodity
Risk, May 1999.
On Insurance Derivatives
Canter, M. S., J. B. Cole, and R. L. Sandor, "Insurance Derivatives: A New
Asset Class for the Capital Markets and a New Hedging Tool for the Insurance Industry," Journal of Applied Corporate Finance, Autumn 1997: 69-83.
I Weather, Energy, and Insurance Derivatives
393
Froot, K. A., "The Market for Catastrophe Risk: A Clinical Examination,"
Journal of Financial Economics, 60 (2001): 529-571.
Froot, K. A., The Financing of Catastrophe Risk. University of Chicago Press,
1999.
Geman, H., "CAT Calls," Risk, September 1994: 86-89.
Hanley, M., "A Catastrophe Too Far," Risk, Supplement on Insurance, July
1998.
Litzenberger, R. H., D. R. Beaglehole, and C. E. Reynolds, "Assessing Catastrophe Reinsurance-Linked Securities as a New Asset Class," Journal of Portfolio Management, Winter 1996: 76-86.
QUESTIONS AND PROBLEMS (Answers at End of Book)
17.1. What is meant by HDD and CDD?
17.2. How is a typical natural gas forward contract structured?
17.3. Suppose that each day during July the minimum temperature is
68° Fahrenheit and the maximum temperature is 82° Fahrenheit. What
is the payoff from a call option on the cumulative CDD during July with
a strike of 250 and a payment rate of $5,000 per degree day?
17.4. Why is the price of electricity more volatile than that of other energy
sources?
17.5. "HDD and CDD can be regarded as payoffs from options on temperature." Explain.
17.6. Suppose that you have 50 years of temperature data at your disposal.
Explain carefully the analyses you would carry out to value a forward
contract on the cumulative CDD for a particular month.
17.7. Would you expect the volatility of the one-year forward price of oil to be
greater than or less than the volatility of the spot price. Explain.
17.8. How can an energy producer use derivative markets to hedge risks?
17.9. Explain how a 5 x 8 option contract on electricity for May 2006 with
daily exercise works. Explain how a 5 x 8 option contract on electricity
for May 2006 with monthly exercise works. Which is worth more?
17.10. Explain how CAT bonds work.
17.11. Consider two bonds that have the same coupon, time to maturity, and
price. One is a B-rated corporate bond. The other is a CAT bond. An
analysis based on historical data shows that the expected losses on the
two bonds in each year of their life is the same. Which bond would you
advise a portfolio manager to buy and why?
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17.12. "Oil, gas, and electricity exhibit mean reversion." What is meant by this
statement? Which product has the highest mean-reversion rate? Which
has the lowest?
ASSIGNMENT QUESTION
17.13. An insurance company's losses of a particular type are to a reasonable
approximation normally distributed with a mean of $150 million and a
standard deviation of $50 million. (Assume no difference between losses
in a risk-neutral world and losses in the real world.) The one-year riskfree rate is 5%. Estimate the cost of the following: (a) a contract that will
pay in one-year's time 60% of the insurance company's costs on a pro
rata basis, and (b) a contract that pays $100 million in one-year's time if
losses exceed $200 million.
Big Losses and
What We Can
Learn from Them
Since the mid-1980s there have been some spectacular losses in financial
markets. This chapter explores the lessons we can learn from them and
emphasizes some of the key points made in earlier chapters. The losses
that we will consider are listed in Business Snapshot 18.1.
One remarkable aspect of the list in Business Snapshot 18.1 is the
number of times huge losses were caused by the activities of a single
employee. In 1995, Nick Leeson's trading brought a 200-year-old British
bank, Barings, to its knees; in 1994, Robert Citron's trading led to Orange
County, a municipality in California, losing about $2 billion. Joseph Jett's
trading for Kidder Peabody caused losses of $350 million. John Rusnak's
losses of $700 million at Allied Irish Bank came to light in 2002.
Some of the losses involve derivatives, but they should not be viewed as
an indictment of the whole derivatives industry. The derivatives market is a
vast multitrillion-dollar market that by most measures has been outstandingly successful and has served the needs of its users well. Derivatives
trades involving unacceptable risks represent a tiny proportion of total
trades (both in number and in value). To quote Alan Greenspan, who was
Chairman of the Federal Reserve, in May 2003:
The use of a growing array of derivatives and the related application of
more-sophisticated methods for measuring and managing risk are key
factors underpinning the enhanced resilience of our largest financial
intermediaries.
Chapter 1
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Business Snapshot 18.1
Big Losses
Allied Irish Bank
This bank lost about $700 million from the unauthorized speculative activities
of one of its foreign exchange traders, John Rusnak, that lasted a number of
years. Rusnak covered up his losses by creating fictitious options trades.
Barings (See Business Snapshot 2.4)
This 200-year-old British bank was wiped out in 1995 by the activities of one
trader, Nick Leeson, in Singapore. The trader's mandate was to arbitrage
between Nikkei 225 futures quotes in Singapore and Osaka. Instead he made
big bets on the future direction of the Nikkei 225 using futures and options.
The total loss was close to $1 billion.
Enron's Counterparties
Enron managed to conceal its true situation from its shareholders with some
creative contracts. Several financial institutions that allegedly helped Enron do
this have had to settle shareholder lawsuits for over $1 billion.
Hammersmith and Fulham (See Business Snapshot 14.1)
This British Local Authority lost about $600 million on sterling interest rate
swaps and options in 1988. The two traders responsible for the loss knew
surprisingly little about the products they were trading.
Kidder Peabody (See Business Snapshot 15.1)
The activities of a single trader, Joseph Jett, led to this New York investment
dealer losing $350 million trading US government securities. The loss arose
because of a mistake in the way the company's computer system calculated
profits.
Long-Term Capital Management (See Business Snapshot 12.1)
This hedge fund lost about $4 billion in 1998 carrying out convergence
arbitrage strategies. The loss was caused by a flight to quality after Russia
defaulted on its debt.
National Westminster Bank
This British bank lost about $130 million from using an inappropriate model
to value swap options in 1997.
Orange County (See Business Snapshot 4.1)
The activities of the treasurer, Robert Citron, led to this California municipality losing about $2 billion in 1994. The treasurer was using derivatives to
speculate that interest rates would not rise.
Procter and Gamble (See Business Snapshot 2.2)
The treasury department of this large US company lost about $90 million in
1994 trading highly exotic interest rate derivatives contracts with Bankers
Trust. It later sued Bankers Trust and settled out of court.
Big Losses and What We Can Learn from Them
397
18.1 RISK LIMITS
The first and most important lesson from the losses concerns risk limits.
It is essential that all companies (financial and nonfinancial) define in a
clear and unambiguous way limits to the financial risks that can be taken.
They should then set up procedures for ensuring that the limits are
obeyed. Ideally, overall risk limits should be set at board level. These
should then be converted to limits applicable to the individuals responsible for managing particular risks. Daily reports should indicate the gain
or loss that will be experienced for particular movements in market
variables. These should be checked against the actual gains and losses
that are experienced to ensure that the valuation procedures underlying
the reports are accurate.
It is particularly important that companies monitor risks carefully
when derivatives are used. This is because derivatives can be used for
hedging or speculation or arbitrage. Without close monitoring, it is
impossible to know whether a derivatives trader has switched from being
a hedger to a speculator or from being an arbitrageur to a speculator.
Barings is a classic example of what can go wrong. Nick Leeson's
mandate was to carry out low-risk arbitrage between the Singapore and
Osaka markets on Nikkei 225 futures. Unknown to his superiors in
London, Leeson switched from being an arbitrageur to taking huge bets
on the future direction of the Nikkei 225. Systems within Barings were so
inadequate that nobody knew what he was doing.
The argument here is not that no risks should be taken. A treasurer
working for a corporation or a trader in a financial institution or a fund
manager should be allowed to take positions on the future direction of
relevant market variables. What we are arguing is that the sizes of the
positions that can be taken should be limited and the systems in place
should accurately report the risks being taken.
A Difficult Situation
What happens if an individual exceeds risk limits and makes a profit?
This is a tricky issue for senior management. It is tempting to ignore
violations of risk limits when profits result. However, this is shortsighted.
It leads to a culture where risk limits are not taken seriously, and it paves
the way for a disaster. The classic example here is Orange County. Robert
Citron's activities in 1991-1993 had been very profitable for Orange
County, and the municipality had come to rely on his trading for
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additional funding. People chose to ignore the risks he was taking because
he had produced profits. Unfortunately, the losses made in 1994 far
exceeded the profits from previous years.
The penalties for exceeding risk limits should be just as great when
profits result as when losses result. Otherwise, traders that make losses are
liable to keep increasing their bets in the hope that eventually a profit will
result and all will be forgiven.
Do Not Assume You Can Outguess the Market
Some traders are quite possibly better than others. But no trader gets it
right all the time. A trader who correctly predicts the direction in which
market variables will move 60% of the time is doing well. If a trader has
an outstanding track record (as Robert Citron did in the early 1990s), it is
likely to be a result of luck rather than superior trading skill.
Suppose that a financial institution employs 16 traders and one of
those traders makes profits in every quarter of a year. Should the trader
receive a good bonus? Should the trader's risk limits be increased? The
answer to the first question is that inevitably the trader will receive a good
bonus. The answer to the second question should be no. The chance of
making a profit in four consecutive quarters from random trading is 0.5
or one in 16. This means that just by chance one of the 16 traders will
"get it right" every single quarter of the year. We should not assume that
the trader's luck will continue and we should not increase the trader's risk
limits.
Do Not Underestimate the Benefits of Diversification
When a trader appears good at predicting a particular market variable,
there is a tendency to increase the trader's risk limits. We have just argued
that this is a bad idea because it is quite likely that the trader has been
lucky rather than clever. However, let us suppose that we are really
convinced that the trader has special talents. How undiversified should
we allow ourselves to become in order to take advantage of the trader's
special skills? The answer, as indicated in Section 1.1, is that the benefits
from diversification are huge, and it is unlikely that any trader is so good
that it is worth foregoing these benefits to speculate heavily on just one
market variable.
An example will illustrate the point here. Suppose that there are
20 stocks, each of which have an expected return of 10% per annum
and a standard deviation of returns of 30%. The correlation between the
Big Losses and What We Can Learn from Them
399
returns from any two of the stocks is 0.2. By dividing an investment
equally among the 20 stocks, an investor has an expected return of
10% per annum and standard deviation of returns of 14.7%. Diversification enables the investor to reduce risks by over half. Another way of
expressing this is that diversification enables an investor to double the
expected return per unit of risk taken. The investor would have to be
extremely good at stock picking to get a better risk-return trade-off by
investing in just one stock.
Carry out Scenario Analyses and Stress Tests
The calculation of risk measures such as VaR should always be accompanied by scenario analyses and stress testing to obtain an understanding
of what can go wrong. These techniques were mentioned in Section 8.7.
They are very important. Human beings have an unfortunate tendency to
anchor on one or two scenarios when evaluating decisions. In 1993 and
1994, for example, Procter and Gamble was so convinced that interest
rates would remain low that, in their decision-making, they ignored the
possibility of a 100 basis point rate increase.
It is important to be creative in the way scenarios are generated. One
approach is to look at ten or twenty years of data and choose the most
extreme events as scenarios. Sometimes there is a shortage of data on a
key variable. It is then sensible to choose a similar variable for which
much more data is available and use historical daily percentage
changes in that variable as a proxy for possible daily percentage
changes in the key variable. For example, if there is little data on
the prices of bonds issued by a particular country, we can look at
historical data on prices of bonds issued by other similar countries to
develop possible scenarios.
18.2 MANAGING THE TRADING ROOM
In trading rooms there is a tendency to regard high-performing traders as
"untouchable" and to not subject their activities to the same scrutiny as
other traders. Apparently Joseph Jett, Kidder Peabody's star trader of
Treasury instruments, was often "too busy" to answer questions and
discuss his positions with the company's risk managers.
It is important that all traders—particularly those making high
profits—be fully accountable. It is important for the financial institution
to know whether the high profits are being made by taking unreasonably
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high risks. It is also important to check that the financial institution's
computer systems and pricing models are correct and are not being
manipulated in some way.
Separate the Front, Middle, and Back Office
The front office in a financial institution consists of the traders who are
executing trades, taking positions, etc. The middle office consists of risk
managers who are monitoring the risks being taken. The back office is
where the record-keeping and accounting takes place. Some of the worst
derivatives disasters have occurred because these functions were not kept
separate. Nick Leeson controlled both the front and back office for
Barings in Singapore and was, as a result, able to conceal the disastrous
nature of his trades from his superiors in London for some time.
Do Not Blindly Trust Models
We discussed model risk in Chapter 15. Some of the large losses experienced by financial institutions arose because of the models and computer
systems being used. Kidder Peabody was misled by its own systems.
Another example of an incorrect model leading to losses is provided by
National Westminster Bank. This bank had an incorrect model for
valuing swap options that led to significant losses.
If large profits are reported when relatively simple trading strategies
are followed, there is a good chance that the models underlying the
calculation of the profits are wrong. Similarly, if a financial institution
appears to be particularly competitive on its quotes for a particular type
of deal, there is a good chance that it is using a different model from
other market participants, and it should analyze what is going on
carefully. To the head of a trading room, getting too much business
of a certain type can be just as worrisome as getting too little business
of that type.
Be Conservative in Recognizing Inception Profits
When a financial institution sells a highly exotic instrument to a nonfinancial corporation, the valuation can be highly dependent on the
underlying model. For example, instruments with long-dated embedded
interest rate options can be highly dependent on the interest rate model
used. In these circumstances, a phrase used to describe the daily marking
to market of the deal is marking to model. This is because there are no
market prices for similar deals that can be used as a benchmark.
Big Losses and What We Can Learn from Them
401
Suppose that a financial institution manages to sell an instrument to a
client for $10 million more than it is worth—or at least $10 million more
than its model says it is worth. The $10 million is known as an inception
profit. When should it be recognized? There appears to be a lot of
variation in what different investment banks do. Some recognize the
$10 million immediately, whereas others are much more conservative
and recognize it slowly over the life of the deal.
Recognizing inception profits immediately is very dangerous. It
encourages traders to use aggressive models, take their bonuses, and
leave before the model and the value of the deal come under close
scrutiny. It is much better to recognize inception profits slowly so that
traders are motivated to investigate the impact of several different models
and several different sets of assumptions before committing themselves to
a deal.
Do Not Sell Clients Inappropriate Products
It is tempting to sell corporate clients inappropriate products, particularly
when they appear to have an appetite for the underlying risks. But this is
shortsighted. The most dramatic example of this is provided by the
activities of Bankers Trust (BT) in the period leading up to the spring
of 1994. Many of BT's clients were persuaded to buy high-risk and totally
inappropriate products. A typical product would give the client a good
chance of saving a few basis points on its borrowings and a small chance
of costing a large amount of money. The products worked well for BT's
clients in 1992 and 1993, but blew up in 1994 when interest rates rose
sharply. The bad publicity that followed hurt BT greatly. The years it had
spent building up trust among corporate clients and developing an
enviable reputation for innovation in derivatives were largely lost as a
result of the activities of a few overly aggressive salesmen. BT was forced
to pay large amounts of money to its clients to settle lawsuits out of
court. It was taken over by Deutsche Bank in 1999.
Enron provides another example of how overly aggressive deal makers
cost their banks billions of dollars. One lesson from Enron is: "The fact
that many banks are pushing hard to get a certain type of business should
not be taken as an indication that the business will be ultimately profitable." Businesses where high profits seem easy to achieve should be
looked at closely for potential operational, credit, or market risks. A
number of banks have had to settle lawsuits with Enron shareholders for
over $1 billion.
402
Chapter 18
18.3 LIQUIDITY RISK
We discussed liquidity risk in Chapter 15. Financial engineers usually
base the pricing of exotic instruments and other instruments that trade
relatively infrequently on the prices of actively traded instruments. For
example:
1. A financial engineer often calculates a zero curve from actively
traded government bonds (known as on-the-run bonds) and uses it
to price bonds that trade less frequently (off-the-run bonds).
2. A financial engineer often implies the volatility of an asset from
actively traded options and uses it to price less actively traded
options.
3. A financial engineer often implies information about the behavior of
interest rates from actively traded interest rate caps and swap
options and uses it to price products that are highly structured.
These practices are not unreasonable. However, it is dangerous to assume
that less actively traded instruments can always be traded at close to their
theoretical price. When financial markets experience a shock of one sort
or another, liquidity black holes may develop (see Section 15.8). Liquidity
then becomes very important to investors, and illiquid instruments often
sell at a big discount to their theoretical values. Trading strategies that
assume large volumes of relatively illiquid instruments can be sold at
short notice at close to their theoretical values are dangerous.
An example of liquidity risk is provided by Long-Term Capital Man
agement (LTCM), which we discussed in Business Snapshot 12.1. This
hedge fund followed a strategy known as convergence arbitrage. It at
tempted to identify two securities (or portfolios of securities) that should
in theory sell for the same price. If the market price of one security was
less that of the other, it would buy that security and sell the other. The
strategy is based on the idea that if two securities have the same
theoretical price their market prices should eventually be the same.
In the summer of 1998 LTCM made a huge loss. This was largely
because a default by Russia on its debt caused a flight to quality. LTCM
tended to be long illiquid instruments and short the corresponding liquid
instruments. (For example, it was long off-the-run bonds and short on-therun bonds.) The spreads between the prices of illiquid instruments and the
corresponding liquid instruments widened sharply after the Russian de
fault. LTCM was highly leveraged. It experienced huge losses and there
were margin calls on its positions that it was unable to meet.
Big Losses and What We Can Learn from Them
403
The LTCM story reinforces the importance of carrying out scenario
analyses and stress testing to look at what can happen in the worst of all
worlds. LTCM could have tried to examine other times in history when
there have been extreme flights to quality to quantify the liquidity risks it
was facing.
Beware When Everyone Is Following the Same Trading Strategy
It sometimes happens that many market participants are following essentially the same trading strategy. This creates a dangerous environment
where there are liable to be big market moves, liquidity black holes, and
large losses for the market participants.
We gave one example of this in Business Snapshot 15.4 when discussing
portfolio insurance and the market crash of October 1987. In the months
leading up to the crash, increasing numbers of portfolio managers were
attempting to insure their portfolios by creating synthetic put options.
They bought stocks or stock index futures after a rise in the market and
sold them after a fall. This created an unstable market. A relatively small
decline in stock prices could lead to a wave of selling by portfolio
insurers. The latter would lead to a further decline in the market, which
could give rise to another wave of selling, and so on. There is little doubt
that without portfolio insurance the crash of October 1987 would have
been much less severe.
Another example is provided by LTCM in 1998. Its position was made
more difficult by the fact that many other hedge funds were following
similar convergence arbitrage strategies. After the Russian default and the
flight to quality, LTCM tried to liquidate part of its portfolio to meet
margin calls. Unfortunately, other hedge funds were facing similar problems to LTCM and trying to do similar trades. This exacerbated the
situation, causing liquidity spreads to be even higher than they would
otherwise have been and reinforcing the flight to quality. Consider, for
example, LTCM's position in US Treasury bonds. It was long the illiquid
off-the-run bonds and short the liquid on-the-run bonds. When a flight to
quality caused spreads between yields on the two types of bonds to widen,
LTCM had to liquidate its positions by selling off-the-run bonds and
buying on-the-run bonds. Other large hedge funds were doing the same.
As a result, the price of on-the-run bonds rose relative to off-the-run
bonds and the spread between the two yields widened even more than it
had already.
A further example is provided by British insurance companies in the
404
Chapter 18
late 1990s. This is discussed in Business Snapshot 15.5. All insurance
companies decided to hedge their exposure to a fall in long-term rates at
about the same time. The result was a fall in long-term rates!
The chief lesson to be learned from these stories is that it is important
to see the big picture of what is going on in financial markets and to
understand the risks inherent in situations where many market participants are following the same trading strategy.
18.4 LESSONS FOR NONFINANCIAL CORPORATIONS
We conclude with some lessons applicable primarily to nonfinancial
corporations.
Make Sure You Fully Understand the Trades You Are Doing
Corporations should never undertake a trade or a trading strategy that
they do not fully understand. This is a somewhat obvious point, but it is
surprising how often a trader working for a nonfinancial corporation will,
after a big loss, admit to not really understanding what was going on and
claim to have been misled by investment bankers. Robert Citron, the
treasurer of Orange County, did this. So did the traders working for
Hammersmith and Fulham, who in spite of their huge positions were
surprisingly uninformed about how the swaps and other interest rate
derivatives they traded really worked.
If a senior manager in a corporation does not understand a trade
proposed by a subordinate, the trade should not be approved. A simple
rule of thumb is that if a trade and the rationale for entering into it are
so complicated that they cannot be understood by the manager, it is
almost certainly inappropriate for the corporation. The trades undertaken by Procter and Gamble would have been vetoed using this
criterion.
One way of ensuring that you fully understand a financial instrument is
to value it. If a corporation does not have the in-house capability to value
an instrument, it should not trade it. In practice, corporations often rely
on their investment bankers for valuation advice. This is dangerous, as
Procter and Gamble found out. When it wanted to unwind its transactions, it found it was facing prices produced by Bankers Trust's
proprietary models, which it had no way of checking.
Big Losses and What We Can Learn from Them
405
Make Sure a Hedger Does Not Become a Speculator
One of the unfortunate facts of life is that hedging is relatively dull,
whereas speculation is exciting. When a company hires a trader to
manage foreign exchange, commodity price, or interest rate risk there is
a danger that the following happens. At first the trader does the job
diligently and earns the confidence of top management. He or she
assesses the company's exposures and hedges them. As time goes by,
the trader becomes convinced that he or she can outguess the market.
Slowly the trader becomes a speculator. At first things go well, but then a
loss is made. To recover the loss, the trader doubles up the bets. Further
losses are made, and so on. The result is likely to be a disaster.
As mentioned earlier, clear limits to the risks that can be taken should
be set by senior management. Controls should be put in place to ensure
that the limits are obeyed. The trading strategy for a corporation should
start with an analysis of the risks facing the corporation in foreign
exchange, interest rate, commodity markets, and so on. A decision
should then be taken on how the risks are to be reduced to acceptable
levels. It is a clear sign that something is wrong within a corporation if
the trading strategy is not derived in a very direct way from the
company's exposures.
Be Cautious about Making the Treasury Department a
Profit Center
In the last 20 years there has been a tendency to make the treasury
department within a corporation a profit center. This seems to have much
to recommend it. The treasurer is motivated to reduce financing costs and
manage risks as profitably as possible. The problem is that the potential
for the treasurer to make profits is limited. When raising funds and
investing surplus cash, the treasurer is facing an efficient market. The
treasurer can usually improve the bottom line only by taking additional
risks. The company's hedging program gives the treasurer some scope for
making shrewd decisions that increase profits, but it should be remembered that the goal of a hedging program is to reduce risks, not to
increase expected profits. The decision to hedge will lead to a worse
outcome than the decision not to hedge roughly 50% of the time. The
danger of making the treasury department a profit center is that the
treasurer is motivated to become a speculator. An outcome like that of
Orange County or Procter and Gamble is then liable to occur.
406
Chapter 18
SUMMARY
The key lesson to be learned from the losses is the importance of internal
controls. The risks taken by traders, the models used, and the amount of
different types of business done should all be controlled. It is important
to "think outside the box" about what could go wrong. LTCM, Enron's
bank counterparties, and many other financial institutions have failed to
do this, with huge adverse financial consequences.
FURTHER READING
Dunbar, N., Inventing Money: The Story of Long-Term Capital Management and
the Legends Behind It. Chichester, UK: Wiley, 2000.
Jorion, P., Big Bets Gone Bad: Derivatives and Bankruptcy in Orange County.
New York: Academic Press, 1995.
Jorion, P., "How Long-Term Lost Its Capital," Risk, September 1999: 31-36.
Ju, X., and N. Pearson, "Using Value at Risk to Control Risk Taking: How
Wrong Can You Be?" Journal of Risk, 1 (1999): 5-36.
Thomson, R., Apocalypse Roulette: The Lethal World of Derivatives. London:
Macmillan, 1998.
Zhang, P. G., Barings Bankruptcy and Financial Derivatives. Singapore: World
Scientific Publishing, 1995.
Valuing Forward
and Futures
Contracts
The forward or futures price of an investment asset that provides no
income is given by
where S0 is the spot price of the asset today, T is the time to maturity of
the forward or futures contract, and r is the continuously compounded
risk-free rate for maturity T. When the asset provides income during the
life of the contract that has a present value I, this becomes
When it provides a yield at rate q, it becomes
A foreign currency can be regarded as an investment asset that provides a
yield equal to the foreign risk-free rate.
The value of a forward contract where the holder has the right to buy
the asset for a price of K is in all cases
where F is the forward price. The value of a forward contract where the
408
Appendix A
holder has the right to sell the asset for a price of K is similarly
Example A.1
Consider a six-month futures contract on the S&P 500. The current value of
the index is 1200, the six-month risk-free rate is 5% per annum, and the
average dividend yield on the S&P 500 over the next six months is expected to
be 2% per annum (both rates continuously compounded). The futures price is
1200e(0.05 - 0.02)x0.5 , or 1,218.14.
Example A.2
The current forward price of gold for a contract maturing in nine months is
$550. A company has a forward contract to buy 1,000 ounces of gold for a
delivery price of $530 in nine months. The nine-month risk-free rate is 4% per
annum continuously compounded. The value of the forward contract is
1,000 x (550 - 530)e -0.04x9/12 , or $19,409.
Valuing Swaps
An interest rate swap can be valued by assuming that the interest rates
that are realized in the future equal today's forward interest rates. As an
example, consider an interest rate swap has 14 months remaining and a
notional principal of $100 million. A fixed rate of 5% per annum is
received and LIBOR is paid, with exchanges taking place every six
months. Assume that (a) four months ago the six-month LIBOR rate
was 4%, (b) the forward LIBOR interest rate for a six-month period
starting in two months is 4.6%, and (c) the forward LIBOR for a sixmonth period starting in eight months is 5.2%. All rates are expressed
with semiannual compounding. Assuming that forward rates are realized,
the cash flows on the swap are as shown in Table B.l. (For example, in
eight months the fixed-rate cash flow received is 0.5 x 0.05 x 100, or
$2.5 million; the floating-rate cash flow paid is 0.5 x 0.046 x 100, or
Table B.1
Valuing an interest rate swap by assuming forward
rates are realized.
Time
Fixed cash flow
($ million)
Floating cash flow
($ million)
Net cash flow
($ million)
2.5
2.5
2.5
-2.0
-2.3
-2.6
0.5
0.2
-0.1
2 months
8 months
14 months
410
Appendix B
Table B.2
Valuing a currency swap by assuming forward exchange rates are
realized (all cash flows in millions).
Time
USD
cash flow
GBP
cash flow
Forward
exchange rate
USD value of
GBP cash flow
Net cash flow
in USD
1
-0.6
-0.6
-0.6
-10.0
0.2
0.2
0.2
5.0
1.8000
1.8400
1.8800
1.8800
0.360
0.368
0.376
9.400
-0.240
-0.232
-0.224
-0.600
2
3
3
$2.3 million.) The value of the swap is the present value of the net cash
flows in the final column.1
An alternative approach (which gives the same valuation) is to assume
that the swap principal of $100 million is paid and received at the end of
the life of the swap. This makes no difference to the value of the swap but
allows it to be regarded as the exchange of interest and principal on a
fixed-rate bond for interest and principal on a floating-rate bond. The
fixed-rate bond's cash flows can be valued in the usual way. A general rule
is that the floating-rate bond is always worth an amount equal to the
principal immediately after an interest payment. In our example, the value
of the floating rate bond is worth $100 million immediately after the
payment in two months. This payment (determined four months ago) is
$2 million. The value of the floating-rate bond is therefore $102 million
immediately before the payment at the two-month point. The value of the
swap is therefore the present value of the fixed-rate bond less the present
value of a cash flow of $102 million in two months.
Currency Swaps
A currency swap can be valued by assuming that exchange rates in the
future equal today's forward exchange rates. As an example consider a
currency swap in which 4% will be received in GBP and 6% will be paid
in USD once a year. The principals in the two currencies are 10 million
USD and 5 million GBP. The swap will last for another three years. The
swap cash flows are shown in the second and third columns of Table B.2.
The forward exchange rates are (we assume) those shown in the fourth
column. These are used to convert the GBP cash flows to USD. The final
1
Note that this is not perfectly accurate because it does not take account of day count
conventions and holiday calendars.
Valuing Swaps
411
column shows the net cash flows. The value of the swap is the present
value of these cash flows.
An alternative approach (which gives the same valuation) is to regard
the swap as a long position in a GBP bond and a short position in a USD
bond. Each bond can be valued in its own currency in the usual way and
the current exchange rate can be used to convert the value of the GBP
bond from GBP to USD.
Valuing
European Options
The Black-Scholes-Merton formulas for valuing European call and put
options on an investment asset that provides no income are
and
where
The function N(x) is the cumulative probability distribution function for
a standardized normal distribution (see tables at the end of the book).
The variables c and p are the European call and European put price, S0 is
the stock price at time zero, K is the strike price, r is the continuously
compounded risk-free rate, is the stock price volatility, and T is the time
to maturity of the option.
When the underlying asset provides a cash income, the present value of
the income during the life of the option should be subtracted from S0.
When the underlying asset provides a yield at rate q, the formulas become
414
Table C.1
Appendix C
Greek letters for options on an asset that provides a yield at rate q.
and
where
Options on a foreign currency can be valued by setting q equal to the
foreign risk-free rate.
Table C.l gives formulas for the Greek letters. N'(x) is the standard
normal density function, given by
Example C.1
Consider a six-month European option on a stock index. The current value of
the index is 1200, the strike price is 1250, the risk-free rate is 5%, the dividend
yield on the index is 2%, and the index volatility is 20%. In this case,
S0 = 1200, K = 1250, r = 0.05, q = 0.02, = 0.2, and T = 0.5. The value of
Valuing
European
Options
415
the option is 53.44, the delta of the option is 0.45, the gamma is 0.0023, the
theta is -0.22, the vega is 3.33, and rho is 2.44. Note that the formula in
Table C.l gives theta per year. The theta quoted here is per calendar day.
The calculations in this appendix can be done with the DerivaGem
software on the author's website by selecting Analytic European for the
Option Type. Option valuation is described more fully in Hull (2006).1
1
See J. C. Hull, Options, Futures, and Other Derivatives, 6th edn., Prentice Hall, 2006.
Valuing
American Options
To value American-style options, we divide the life of the option into n
time steps of length
Suppose that the asset price at the beginning of a
step is S. At the end of the time step it moves up to Su with probability p
and down to Sd with probability 1 — p. For an investment asset that
provides no income, the values of u, d and p are given by
Figure D.l shows the tree constructed for valuing a five-month American
put option on a non-dividend-paying stock where the initial stock price is
50, the strike price is 50, the risk-free rate is 10%, and the volatility is
40%. In this case, there are five steps, so that
= 0.08333, u = 1.1224,
d = 0.8909, a — 1.0084, and p = 0.5073. The upper number at each node
is the stock price and the lower number is the value of the option.
At the final nodes of the tree the option price is its intrinsic value. For
example, at node G the option price is 50 — 35.36 = 14.64. At earlier nodes
we first calculate a value assuming that the option is held for a further time
period of length At and then check to see whether early exercise is optimal.
Consider first node E. If the option is held for a further time period it will
be worth 0.00 if there is an up move (probability: p) and 5.45 if there is a
418
Appendix D
At each node:
Upper value = Underlying Asset Price
Lower value = Option Price
Shading indicates where option is exercised
Figure D.1
Binomial tree from DerivaGem for an American put on a
non-dividend-paying stock.
down move (probability: 1 — p). The expected value in time
is therefore
0.5073 x 0 + 0.4927 x 5.45, or 2.686, and the 2.66 value at node E is
calculated by discounting this at the risk-free rate of 10% for one month.
The option should not be exercised at node E as the payoff from early
exercise would be zero. Consider next node A. A similar calculation to that
just given shows that, assuming it is held for a further time period, the
option's value at node A is 9.90. If exercised, its value is 50 — 39.69 =
10.31. In this case, it should be exercised and the value of being at node A
is 10.31.
Continuing to work back from the end of the tree to the beginning, the
value of the option at the initial node D is found to be 4.49. As the number
of steps on the tree is increased, the accuracy of the option price increases.
Valuing
American
Options
419
With 30, 50, and 100 time steps, we get values for the option of 4.263,
4.272, and 4.278.
To calculate delta, we consider the two nodes at time
In our
example, as we move from the lower node to the upper node the option
price changes from 6.96 to 2.16 and the stock price changes from 44.55 to
56.12. The estimate of delta is the change in the option price divided by
the change in the stock price:
To calculate gamma, we consider the three nodes at time
The delta
calculated from the upper two nodes (C and F) is —0.241. This can be
regarded as the delta for a stock price of (62.99 + 50)/2 = 56.49. The
delta calculated from the lower two nodes (B and C) is —0.639. This can
be regarded as the delta for a stock price of (50 + 39.69)/2 = 44.84. The
estimate of gamma is the change in delta divided by the change in the
stock price:
We estimate theta from nodes D and C as
or —4.30 per year. This is -0.0118 per calendar day. Vega is estimated by
increasing the volatility, constructing a new tree, and observing the effect
of the increased volatility on the option price. Rho is calculated similarly.
When the asset underlying the option provides a yield at rate q the
procedure is exactly the same except that a =
instead of
in the
equation for p. The calculations we have described can be done using the
DerivaGem software by selecting Binomial American for the Option
Type. Binomial trees and other numerical procedures are described more
fully in Hull (2006). 1
1
See J. C. Hull, Options, Futures, and Other Derivatives, 6th edn., Prentice Hall, 2006.
The Manipulation
of Credit Transition
Matrices
Suppose that A is an N x N matrix of credit rating changes in one year.
This is a matrix such as the one shown in Table 12.1. The matrix of credit
rating changes in m years is Am. This can be readily calculated using the
normal rules for matrix multiplication.
The matrix corresponding to a shorter period than one year, say six
months or one month, is more difficult to compute. We first use standard
routines to calculate eigenvectors
and the corresponding
eigenvalues
These have the property that
Define X as a matrix whose ith row is
and
a s a diagonal matrix
where the ith diagonal element is
A standard result in matrix algebra
shows that
From this it is easy to see that the nth root of A is
where
is a diagonal matrix where the ith diagonal element is
Some authors, such as Jarrow, Lando, and Turnbull,1 prefer to handle
1
See R. A. Jarrow, D. Lando, and S. M. Turnbull, "A Markov Model for the Term
Structure of Credit Spreads," Review of Financial Studies, 10 (1997), 481-523.
422
Appendix E
this problem in terms of what is termed a generator matrix. This is a
matrix
such that the transition matrix for a short period of time
is
I+
where I is the identity matrix, and the transition matrix for a
longer period of time t is
Answers to Questions
and Problems
CHAPTER 1
1.1. Expected return is 12.5%. SD of return is 17.07%.
1.2. From equations (1.1) and 1.2), expected return is 12.5%. SD of
return is
or 12.94%.
1.3.
( =0.3)
0.0
0.2
0.4
0.6
0.8
1.0
1.0
0.8
0.6
0.4
0.2
0.0
15%
14%
13%
12%
11%
10%
24.00%
20.39%
17.42%
15.48%
14.96%
16.00%
( =1)
24.00%
22.40%
20.80%
19.20%
17.60%
16.00%
(
=-1)
24.00%
16.00%
8.00%
0.00%
8.00%
16.00%
1.4. Nonsystematic risk can be diversified; systematic risk cannot. Systematic risk is most important to an equity investor. Either type of risk
can lead to the bankruptcy of a corporation.
1.5. We assume that investors trade off mean return and standard
deviation of return. For a given mean return, they want to minimize
424
Answers to Problems and Questions
standard deviation of returns. All make the same estimates of means
standard deviations, and coefficients of correlation for returns on individual investments. Furthermore they can borrow or lend at the risk-free
rate. The result is that they all want to be on the "new efficient frontier"
in Figure 1.4. They choose the same portfolio of risky investments
combined with borrowing or lending at the risk-free rate.
1.6. (a) 7.2%, (b) 9%, (c) 14.4%.
1.7. The capital asset pricing theory assumes that there is one factor
driving returns. Arbitrage pricing theory assumes multiple factors.
1.8. In many jurisdictions, interest on debt is deductible to the corporation
whereas dividends are not deductible. It can therefore be more tax efficient
for a company to fund itself with debt. However, as debt increases, the
probability of bankruptcy increases.
1.9. When potential losses are large, we cannot aggregate them and
assume they will be diversified away. It is necessary to consider them
one by one and handle them with insurance contracts, tighter internal
controls, etc.
1.10. This is the probability that profit is no worse than —4% of assets.
This profit level is 4.6/1.5 = 3.067 standard deviations from the mean. The
probability that the bank will have a positive equity is therefore N(3.067),
where N is the cumulative normal distribution function. This is 99.89%.
1.11. Banks have the privilege of being allowed to take money from
depositors. Companies in retailing and manufacturing do not.
1.12. There was an interest rate mismatch at Continental Illinois. About
$5.5 billion of loans with maturities more than a year were financed by
deposits with maturities less than a year. If interest rates rose 1%, the
deposits would be rolled over at higher rates while the loans would
continue to earn the same rate. The cost to Continental Illinois would
be $55 million.
1.13. S&Ls financed long-term fixed-rate mortgages with short-term
deposits creating a serious interest rate mismatch. As a result, they lost
money when interest rates rose.
1.14. In this case, the interest rate mismatch is $10 billion. The bank's net
interest income declines $100 million each year for the next three years.
1.15. Professional fees ($5 million per month), lost sales (people are
reluctant to do business with a company that is being reorganized), and
key senior executives left (lack of continuity).
Answers to Problems and Questions
425
CHAPTER 2
2.1. When a trader enters into a long forward contract, she is agreeing to
buy the underlying asset for a certain price at a certain time in the future.
When a trader enters into a short forward contract, she is agreeing to sell
the underlying asset for a certain price at a certain time in the future.
2.2. A trader is hedging when she has an exposure to the price of an asset
and takes a position in a derivative to offset the exposure. In a speculation
the trader has no exposure to offset. She is betting on the future movements in the price of the asset. Arbitrage involves taking a position in two
or more different markets to lock in a profit.
2.3. In the first case, the trader is obligated to buy the asset for $50 (she
does not have a choice). In the second case, the trader has an option to
buy the asset for $50 (she does not have to exercise the option).
2.4. Selling a call option involves giving someone else the right to buy an
asset from you for a certain price. Buying a put option gives you the right
to sell the asset to someone else.
2.5. (a) The investor is obligated to sell pounds for 1.5000 when they are
worth 1.4900. The gain is (1.5000 - 1.4900) x 100,000 = $1,000. (b) The
investor is obligated to sell pounds for 1.5000 when they are worth 1.5200.
The loss is (1.5200 - 1.5000) x 100,000 = $2,000.
2.6. (a) The trader sells for 50 cents per pound something that is worth
48.20 cents per pound. Gain = ($0.5000 - $0.4820) x 50,000 = $900.
(b) The trader sells for 50 cents per pound something that is worth 51.30
cents per pound. Loss = ($0.5130 - $0.5000) x 50,000 = $650.
2.7. You have sold a put option. You have agreed to buy 100 shares for
$40 per share if the party on the other side of the contract chooses to
exercise the right to sell for this price. The option will be exercised only
when the price of stock is below $40. Suppose, for example, that the
option is exercised when the price is $30. You have to buy at $40 shares
that are worth $30; you lose $10 per share, or $1,000 in total. If the
option is exercised when the price is $20, you lose $20 per share, or $2,000
in total. The worst that can happen is that the price of the stock declines
to almost zero during the three-month period. This highly unlikely event
would cost you $4,000. In return for the possible future losses, you receive
the price of the option from the purchaser.
2.8. The over-the-counter (OTC) market is a telephone- and computerlinked network of financial institutions, fund managers, and corporate
treasurers where two participants can enter into any mutually acceptable
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contract. An exchange-traded market is a market organized by an exchange where traders either meet physically or communicate electronically
and the contracts that can be traded have been defined by the exchange
(a) OTC, (b) exchange, (c) both, (d) OTC, (e) OTC.
2.9. One strategy would be to buy 200 shares. Another would be to buy
2,000 options. If the share price does well, the second strategy will give
rise to greater gains. For example, if the share price goes up to $40, you
gain [2, 000 x ($40 - $30)] - $5,800 = $14,200 from the second strategy
and only 200 x ($40 - $29) = $2,200 from the first. However, if the
share price does badly, the second strategy gives greater losses. For
example, if the share price goes down to $25, the first strategy leads
to a loss of 200 x ($29 — $25) = $800, whereas the second strategy leads
to a loss of the whole $5,800 investment. This example shows that
options contain built in leverage.
2.10. You could buy 5,000 put options (or 50 contracts) with a strike
price of $25 and an expiration date in 4 months. This provides a type of
insurance. If at the end of 4 months the stock price proves to be less than
$25, you can exercise the options and sell the shares for $25 each. The
cost of this strategy is the price you pay for the put options.
2.11. A stock option provides no funds for the company. It is a security
sold by one trader to another. The company is not involved. By contrast,
a stock when it is first issued is a claim sold by the company to investors
and does provide funds for the company.
2.12. Ignoring the time value of money, the holder of the option will
make a profit if the stock price in March is greater than $52.50. This is
because the payoff to the holder of the option is, in these circumstances,
greater than the $2.50 paid for the option. The option will be exercised if
the stock price at maturity is greater than $50.00. Note that if the stock
price is between $50.00 and $52.50 the option is exercised, but the holder
of the option takes a loss overall.
2.13. Ignoring the time value of money, the seller of the option will make
a profit if the stock price in June is greater than $56.00. This is because
the cost to the seller of the option is in these circumstances less than the
price received for the option. The option will be exercised if the stock
price at maturity is less than $60.00. Note that if the stock price is
between $56.00 and $60.00 then the seller of the option makes a profit
even though the option is exercised.
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427
2.14. A long position in a four-month put option can provide insurance
against the exchange rate falling below the strike price. It ensures that the
foreign currency can be sold for at least the strike price.
2.15. The company could enter into a long forward contract to buy
1 million Canadian dollars in six months. This would have the effect of
locking in an exchange rate equal to the current forward exchange rate.
Alternatively the company could buy a call option giving it the right (but
not the obligation) to purchase 1 million Canadian dollars at a certain
exchange rate in six months. This would provide insurance against a
strong Canadian dollar in six months while still allowing the company
to benefit from a weak Canadian dollar at that time.
2.16. The payoff from an ICON is the payoff from (a) a regular bond,
(b) a short position in call options to buy 169,000 yen with an exercise
price of 1/169, (c) a long position in call options to buy 169,000 yen with
an exercise price of 1/84.5. This is demonstrated by the following table:
2.17. (a) The trader buys a 180-day call option and takes a short position
in a 180-day forward contract. (b) The trader buys 90-day put options
and takes a long position in a 90-day forward contract.
2.18. It enters into a 5-year swap where it pays 6.51% and receives
LIBOR. Its investment is then at LIBOR minus 1.51%.
2.19. It enters into a 5-year swap where it receives 6.47% and pays
LIBOR. Its net cost of borrowing is LIBOR +0.53%.
2.20. It enters into a three-year swap where it receives LIBOR and pays
6.24%. Its net borrowing cost for the three years is then 7.24% per
annum.
2.21. Suppose that the weather is bad and the farmer's production is
lower than expected. Other farmers are likely to have been affected
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similarly. Corn production overall will be low and as a consequence the
price of corn will be relatively high. The farmer is likely to be overhedged
relative to actual production. The farmer's problems arising from the bad
harvest will be made worse by losses on the short futures position. This
problem emphasizes the importance of looking at the big picture when
hedging. The farmer is correct to question whether hedging price risk
while ignoring other risks is a good strategy.
2.22. It may well be true that there is just as much chance that the
company will lose as that it will gain. This means that the use of a futures
contract for speculation would be like betting on whether a coin comes up
heads or tails. But it might make sense for the airline to use futures for
hedging rather than speculation. The futures contract then has the effect
of reducing risks. It can be argued that an airline should not expose its
shareholders to risks associated with the future price of oil when there are
contracts available to hedge the risks.
2.23. The optimal hedge ratio is
The beef producer requires a long position in 200,000 x 0.6 = 120,000 lbs
of cattle. The beef producer should therefore take a long position in three
December contracts closing out the position on November 15.
2.24. Microsoft is choosing an option on a portfolio of assets instead of
the corresponding portfolio of options. The former is always less expensive because there is the potential for an increase in the price of one asset
to be netted off against a decrease in the price of another. Compare (a) an
option with a strike price of $20 on a portfolio of two assets each worth
$10 and (b) a portfolio of two options with a strike price of $10, one on
each of assets. If both assets increase in price or both assets decrease in
price, the payoffs are the same. But if one decreases and the other
increases, the payoff from (a) is less than that from (b). Both the Asian
feature and the basket feature in Microsoft's options help to reduce the
cost of the options because of the possibility of gains and loss being
netted.
CHAPTER 3
3.1. The value of the portfolio decreases by $10,500.
3.2. The value of the portfolio increases by $400.
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429
3.3. In both cases it increases by 0.5 x 30 x 22, or $60.
3.4. A delta of 0.7 means that, when the price of the stock increases by a
small amount, the price of the option increases by 70% of this amount.
Similarly, when the price of the stock decreases by a small amount, the
price of the option decreases by 70% of this amount. A short position in
1,000 options has a delta of —700 and can be made delta neutral with the
purchase of 700 shares.
3.5. A theta of — 100 per day means that if one day passes with no change
in either the stock price or its volatility, the value of the option position
declines by $100. If a trader feels that neither the stock price nor its
implied volatility will change, she should write an option with as high a
theta as possible. Relatively short-life at-the-money options have the
highest theta.
3.6. The gamma of an option position is the rate of change of the delta of
the position with respect to the asset price. For example, a gamma of 0.1
would indicate that, when the asset price increases by a certain small
amount, delta increases by 0.1 of this amount. When the gamma of an
option-writer's position is large and negative and the delta is zero, the
option writer will lose significant amounts of money if there is a large
movement (either an increase or a decrease) in the asset price.
3.7. To hedge an option position, it is necessary to create the opposite
option position synthetically. For example, to hedge a long position in a
put, it is necessary to create a short position in a put synthetically. It
follows that the procedure for creating an option position synthetically is
the reverse of the procedure for hedging the option position.
3.8. A long position in either a put or a call option has a positive gamma.
From Figure 15.8, when gamma is positive the hedger gains from a large
change in the stock price and loses from a small change in the stock price.
Hence the hedger will fare better in case (b). When the portfolio contains
short option position, the hedger will similarly fare better in (a).
3.9. The delta indicates that, when the value of the euro exchange rate
increases by $0.01, the value of the bank's position increases by
0.01 x 30,000 = $300. The gamma indicates that, when the euro exchange
rate increases by $0.01, the delta of the portfolio decreases by
0.01 x 80,000 = 800. For delta neutrality, 30,000 euros should be shorted.
When the exchange rate moves up to 0.93, we expect the delta of the
portfolio to decrease by (0.93 - 0.90) x 80,000 = 2,400, so that it becomes
27,600. To maintain delta neutrality, it is therefore necessary for the bank
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to unwind its short position 2,400 euros so that a net 27,600 have been
shorted. When a portfolio is delta neutral and has a negative gamma, a loss
is experienced when there is a large movement in the underlying asset price
We can conclude that the bank is likely to have lost money.
3.10. When used in the way described in the text, it does assume volatility
is constant. In theory, we could implement a static options replication
scheme where there are three dimensions: time, the stock price, and volatility. Prices are then matched on a surface in the three-dimensional space.
3.11. Ten regular options are likely to be needed. This is because there are
ten equations to be satisfied, one for each point on the boundary.
3.12. The payoff from an Asian option becomes more certain with the
passage of time. As a result, the amount of uncertainty that needs to be
hedged decreases with the passage of time.
3.13. Consider a portfolio of options dependent on a single market
variable. A single trade is all that is necessary to make the position delta
neutral, regardless of the size of the position.
3.14. The price, delta, gamma, vega, theta, and rho are 0.0217, -0.396,
5.415, 0.00203, -0.0000625, and -0.00119. Delta predicts that the option
price should decrease by approximately 0.000396 when the exchange rate
increases by 0.001. This is what we find. When the exchange rate is
increased to 0.751, the option price decreases to 0.0213.
CHAPTER 4
4.1. (a) 13.76% per annum, (b) 14.75% per annum.
4.2. (a) 10% per annum, (b) 9.76% per annum, (c) 9.57%> per annum,
(d) 9.53%o per annum.
4.3. The equivalent rate of interest with quarterly compounding is
12.18%o. The amount of interest paid each quarter is therefore
or $304.55.
4.4. The rate of interest is 14.91%) per annum.
4.5. The forward rates with continuous compounding for the 2nd, 3rd,
4th, and 5th years are 4.0%, 5.1%, 5.7%, and 5.7%, respectively.
4.6. The forward rates with continuous compounding for the 2nd, 3rd,
4th, 5th, and 6th quarters are 8.4%, 8.8%, 8.8%, 9.0%, and 9.2%,
respectively.
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431
4.7. When the term structure is upward sloping, c > a > b. When it is
downward sloping, b > a > c.
4.8. Suppose the bond has a face value of $100. Its price is obtained by
discounting the cash flows at 10.4%. The price is
If the 18-month zero rate is R, we must have
which gives R = 10.42%.
4.9. The bond pays $2 in 6, 12, 18, and 24 months, and $102 in
30 months. The cash price is
4.10. The bond pays $4 in 6, 12, 18, 24, and 30 months, and $104 in
36 months. The bond yield is the value of y that solves
Using the Goal Seek tool in Excel, we get y = 0.06407, or 6.407%.
4.11. There are three reasons: (i) Treasury bills and Treasury bonds must
be purchased by financial institutions to fulfill a variety of regulatory
requirements. This increases demand for these Treasury instruments driving the price up and the yield down. (ii) The amount of capital a bank is
required to hold to support an investment in Treasury bills and bonds is
substantially smaller than the capital required to support a similar investment in other very-low-risk instruments. (iii) In the United States, Treasury instruments are given a favorable tax treatment compared with most
other fixed-income investments because they are not taxed at the state level.
4.12. Duration provides information about the effect of a small parallel
shift in the yield curve on the value of a bond portfolio. The percentage
decrease in the value of the portfolio equals the duration of the portfolio
multiplied by the amount by which interest rates are increased in the
small parallel shift. Its limitation is that it applies only to parallel shifts in
the yield curve that are small.
4.13. (a) The bond's price is 86.80, (b) the bond's duration is 4.256 years,
(c) the duration formula shows that when the yield decreases by 0.2% the
bond's price increases by 0.74, (d) recomputing the bond's price with a
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432
yield of 10.8% gives a price of 87.54, which is approximately consistent
with (a) and (c).
4.14. (a) The bond's price is 88.91, (b) the bond's modified duration is
3.843 years, (c) the duration formula estimates that when the yield
decreases by 0.2% the bond's price increases by 0.68, (d) recomputing
the bond's price with a yield of 10.8% (annually compounded) gives a
price of 89.60 which is approximately consistent with (a) and (c).
4.15. The bond price is 104.80. The duration of the bond is 5.35. The
convexity is 30.60. The effect of a 1 % increase in the yield is estimated by
equation (4.14) as
104.80(-0.01 x 5.35 + 0.5 x 30.60 x 0.0001) = -5.44
The bond price actually changes to 99.36, which is consistent with the
estimate.
4.16. We can (a) perturb points on the yield curve (see Figure 4.4),
(b) perturb sections of the yield curve (see Figure 4.6), and (c) perturb
the market quotes used to create the yield curve.
4.17. The deltas are 10.7 and -190,1.
CHAPTER 5
5.1. 4.16%.
5.2. The standard deviation of the percentage price change in one day is
1.57%. The 95% confidence limits are from -3.09% to +3.09%.
5.3. Volatility is much higher when markets are open than when they are
closed. Traders therefore measure time in trading days rather than
calendar days when applying volatility.
5.4. Implied volatility is the volatility that leads to the option price
equaling the market price when Black-Scholes assumptions are used. It
is found by "trial and error". Because different options have different
implied volatilities, traders are not using the same assumptions as BlackScholes. (See Chapter 15 for a further discussion of this.)
5.5. The approach in Section 5.3 gives 0.547% per day. The simplified
approach in equation 5.4 gives 0.530%) per day.
5.6. (a) 0.25%, (b) 0.0625%.
5.7. The variance rate estimated at the end of day n equals times the
variance rate estimated at the end of day n — 1 plus 1 — times the
squared return on day n.
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5.8. GARCH(1, 1) adapts the EWMA model by giving some weight to a
long-run average variance rate. Whereas the EWMA has no mean reversion, GARCH(1,1) is consistent with a mean-reverting variance rate
model.
5.9. In this case,
= 0.015 and
= 0.5/30 = 0.01667, so that equation (19.7) gives
= 0.94 x 0.0152 + 0.06 x 0.016672 = 0.0002281
The volatility estimate on day n is therefore
1.5103%.
= 0.015103, or
5.10. Reducing from 0.95 to 0.85 means that more weight is put on
recent observations of
and less weight is given to older observations.
Volatilities calculated with
— 0.85 will react more quickly to new
information and will "bounce around" much more than volatilities
calculated with — 0.95.
5.11. With the usual notation,
= 20/1040 = 0.01923, so that
= 0.000002 + 0.06 x 0.019232 + 0.92 x 0.012 = 0.0001162
This gives
= 0.01078. The new volatility estimate is therefore 1.078%
per day.
5.12. The proportional daily change is -0.005/1.5000 = -0.003333. The
current daily variance estimate is 0.006 = 0.000036. The new daily
variance estimate is
0.9 x 0.000036 + 0.1 x 0.0033332 = 0.000033511
The new daily volatility is the square root of this. It is 0.00579, or 0.579%.
5.13. The weight given to the long-run average variance rate is
and the long-run average variance rate is
Increasing
increases the long-run average variance rate; increasing increases the
weight given to the most recent data item, reduces the weight given to the
long-run average variance rate, and increases the level of the long-run
average variance rate. Increasing
increases the weight given to the
previous variance estimate, reduces the weight given to the long-run
average variance rate, and increases the level of the long-run average
variance rate.
5.14. The long-run average variance rate is
or
0.000004/0.03 = 0.0001333. The long-run average volatility is
or 1.155%. The equation describing the way the variance
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434
rate reverts to its long-run average is
In this case,
If the current volatility is 20% per year,
expected variance rate in 20 days is
= 0.0126. The
0.0001333 + 0.9720(0.01262 - 0.0001333) = 0.0001471
The expected volatility in 20 days is therefore
1.21% per day.
=0.0121, or
5.15. The FTSE expressed in dollars is XY where X is the FTSE
expressed in sterling and Y is the exchange rate (value of one pound in
dollars). Define as the proportional change in X on day i and as the
proportional change in Y on day i. The proportional change in XY is
approximately
The standard deviation of
is 0.018 and the
standard deviation of
is 0.009. The correlation between the two is
0.4. The variance of
is therefore
0.0182 + 0.0092 + 2 x 0.018 x 0.009 x 0.4 = 0.0005346
so that the volatility of
is 0.0231, or 2.31 %. This is the volatility of
the FTSE expressed in dollars. Note that it is greater than the volatility of
the FTSE expressed in sterling. This is the impact of the positive correlation. When the FTSE increases, the value of sterling measured in dollars
also tends to increase. This creates an even bigger increase in the value of
FTSE measured in dollars. Similarly for a decrease in the FTSE.
5.16. In this case, VL = 0.00015 and the expected variance rate in 30 days
is 0.000123. The volatility is 1.11% per day.
5.17. In equation (5.15), VL =0.0001, a = 0.0202, T = 20, and V(0) =
0.000169, so that the volatility is 19.88%.
CHAPTER 6
6.1. You need the standard deviations of the two variables.
6.2. Loosely speaking correlation measures the extent of linear dependence. It does not measure other types of dependence. When y = x2 there
is perfect dependence between x and y. However E(xy) = E(x3). This is
zero for a symmetrical distribution such as the normal showing that the
coefficient of correlation between x and y is zero.
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435
6.3. In a factor model the correlation between two variables arises
entirely because of their correlation with one or more other variables.
The latter are known as factors. A factor model reduces the number of
estimates that have to be made when correlations between large numbers
of variables are being produced.
6.4. A Positive-semidefinite matrix is a matrix that satisfies equation (6.4)
for all vectors
If a correlation matrix is not positive semidennite, the
correlations are internally inconsistent.
6.5. (a) The volatilities and correlation imply that the current estimate of
the Covariance is 0.25 x 0.016 x 0.025 = 0.0001. (b) If the prices of the
assets at close of trading are $20.5 and $40.5, the proportional changes
are 0.5/20 = 0.025 and 0.5/40 = 0.0125. The new Covariance estimate is
0.95 x 0.0001 + 0.05 x 0.025 x 0.0125 = 0.0001106
The new variance estimate for asset A is
0.95 x 0.0162 + 0.05 x 0.0252 = 0.00027445
so that the new volatility is 0.0166. The new variance estimate for asset B is
0.95 x 0.0252 + 0.05 x 0.01252 = 0.000601562
so that the new volatility is 0.0245. The new correlation estimate is
6.6. Using the notation in the text,
= 0.01 and
=0.012 and
the most recent estimate of the Covariance between the asset returns is
= 0.01 x 0.012 x 0.50 = 0.00006
The variable
= 1/30 = 0.03333 and the variable
= 1/50 = 0.02.
The new estimate of the Covariance, covn, is
0.000001 + 0.04 x 0.03333 x 0.02 + 0.94 x 0.00006 = 0.0000841
The new estimate of the variance of the first asset,
is
2
2
0.000003 + 0.04 x 0.03333 + 0.94 x 0.01 = 0.0001414
so that
= 0.01189, or 1.189%. The new estimate of the
variance of the second asset,
is
0.000003 + 0.04 x 0.022 + 0.94 x 0.0122 = 0.0001544
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so that ,
= 0.01242, or 1.242%. The new estimate of the
correlation between the assets is therefore
0.0000841/(0.01189 x 0.01242) = 0.569
6.7. Continuing with the notation in Problem 5.15, define as the proportional change in the value of the S&P 500 on day i. The Covariance
between
and
is 0.7 x 0.018 x 0.016 = 0.0002016. The Covariance
between
and
is 0.3 x 0.009 x 0.016 = 0.0000432. The Covariance
between
and
equals the Covariance between
and
plus the
Covariance between and
It is
0.0002016 + 0.0000432 = 0.0002448
The correlation between
and
is
Note that the volatility of the S&P 500 drops out in this calculation.
6.8.
V2
V1
0.25
0.5
0.75
0.25
0.5
0.75
0.095
0.163
0.216
0.163
0.298
0.413
0.216
0.413
0.595
6.9. Suppose x1, x2, and x3 are random samples from three independent
normal distributions. Random samples with the required correlation
structure are
where
where
This means that
6.10. Tail dependence is the tendency for extreme values for two or more
variables to occur together. The choice of copula affects tail dependence.
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437
For example, the Student t-copula gives more tail dependence than the
Gaussian copula.
6.11. Sample from a bivariate Student t-distribution as in Figure 6.5.
Convert each sample to a normal distribution on a "percentile-topercentile" basis.
6.12. The probability that V1 < 0.1 is 0.05. The conditional probability
that V 2 < 0 . 1 is 0.006/0.05 = 0.12. The conditional probability that
V2 < 0.2 is 0.017/0.05 = 0.34, etc.
6.13. When V1= 0.2, we have U1 = —0.84. From the properties of the
bivariate normal distribution, the median of U2 is —0.5 x 0.84 = —0.42.
This translates into a median value for V2 of 0.458.
6.14. In this case,
The "99.5% worst case" is that there is a loss of 500 x 0.7 x 0.127 =
44.62, or $44.62 million.
CHAPTER 7
7.1. The removal of a competitor may be beneficial. However, banks
enter into many contracts with each other. When one bank goes bankrupt, other banks are liable to lose money on the contracts they have with
the bank. Also, other banks will be adversely affected if the bankruptcy
reduces the public's overall level of confidence in the banking system.
7.2. Deposit insurance means that depositors are safe regardless of the
risks taken by their financial institution. It is liable to lead to financial
institutions taking more risks than they otherwise would because they can
do so without the risk of losing deposits. This in turn leads to more bank
failures and more claims under the deposit insurance system. Regulation
requiring the capital held by a bank to be related to the risks taken is
necessary to avoid this happening.
7.3. The credit risk on the swap is the risk that the counterparty defaults
at some future time when the swap has a positive value to the bank.
7.4. The value of a currency swap is liable to deviate further from zero
than the value of an interest rate swap because of the final exchange of
principal. As a result the potential loss from a counterparty default is
higher.
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Answers to Problems and Questions
7.5. There is some exposure. If the counterparty defaulted now there
would be no loss. However, interest rates could change so that at a future
time the swap has a positive value to the financial institution. If the
counterparty defaulted at that time there would be a loss to the financial
institution.
7.6. The risk-weighted assets for the three transactions are (a) $1.875
million, (b) $2 million, and (c) $3 million, for a total of $6.875 million.
Capital is 0.08 x 6.875, or $0.55 million.
7.7. The NRR is 2.5/4.5 = 0.556. The credit equivalent amount is
2.5+ (0.4 +0.6 x 0.556) x 9.25, or $9.28 million. The risk-weighted
assets is $4.64 million and the capital required is $0.371 million.
7.8. In this case there is no value to the netting provisions.
7.9. This converts the estimated capital requirement to an estimated riskweighted assets. Capital required equals 8% of risk-weighted assets.
7.10. The trading book consists of instruments that are actively traded
and marked to market daily. The banking book consists primarily of
loans and is not market to market daily. Prior to the change the bank
keeps credit risk capital calculated according to Basel I or Basel II. The
effect of the change is to move the clients borrowings from the banking
book to the trading book. The bank will be required to hold specific risk
capital for the securities reflecting the credit exposure, as well as market
risk capital reflecting the market risk exposure. The previous credit risk
capital is no longer required.
7.11. Under Basel I the capital charged for lending to a corporation is the
same regardless of the credit rating of the corporation. This leads to a
bank's return on capital being relatively low for lending to highly creditworthy corporations. Under Basel II the capital requirements of a loan are
tied much more carefully to the creditworthiness of the borrower. As a
result lending to highly creditworthy companies may become attractive
again.
7.12. Regulatory arbitrage involves entering into a transaction or series
of transactions solely to reduce regulatory capital requirements.
7.13. EAD is the estimated exposure at default. LGD is the loss given
default, that is, the proportion of the exposure that will be lost if a default
occurs. WCDR is the one-year probability of default in a bad year that
occurs only one time in 1,000. PD is the probability of default in an
average year. MA is the maturity adjustment. The latter allows for the fact
that in the case of instruments lasting longer than a year there may be
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to Problems and Questions
439
losses arising from a decline in the creditworthiness of the counterparty
during the year as well as from a default during the year.
7.14. Under the simple approach, the risk weight of the counterparty is
replaced by the risk weight of the collateral for the part of the exposure
covered by the collateral. Under the comprehensive approach, the exposure
is adjusted for possible increases and the collateral is adjusted for possible
decreases in value. The counterparty's risk weight is applied to the excess
of the adjusted exposure over the adjusted collateral.
7.15. The standardized approach uses external ratings to determine
capital requirements (but in a more sophisticated way than in Basel I).
In the IRB approach the Basel II correlation model is used with PD being
determined by the bank. In the advanced IRB approach, the Basel II
correlation model is used with PD, LGD, EAD, and MA being determined by the bank.
7.16. In the basic indicator approach total capital is 15% of the average
total annual gross income. In the standardized approach, gross income is
calculated for different business lines and capital as a percentage of gross
income is different for different business lines. In the advanced measurement approach, the bank uses internal models to determine the 1-year
99.9% VaR.
7.17. =0.1216, WCDR = 0.0914, and the capital requirement is
200 x 0.7 x 0.0814, or $11.39 million. At least half of this must be Tier I.
CHAPTER 8
8.1. VaR is the loss that is not expected to be exceeded with a certain
confidence level. Expected shortfall is the expected loss conditional that
the loss is worse than the VaR level. Expected shortfall has the
advantage that it always satisfies the Subadditivity (diversification is
good) condition.
8.2. A spectral risk measure is a risk measure that assigns weights to the
quantiles of the loss distribution. For the Subadditivity condition to be
satisfied the weight assigned to the qth quantile must be a nondecreasing
function of q.
8.3. There is a 5% chance that you will lose $6,000 or more during a onemonth period.
8.4. Your expected loss during a "bad month" is $6,000. Bad months are
defined as the worst 5% of months.
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Answers to Problems and Questions
8.5. (a) $1 million, (b) $9.1 million, (c) $11 million, (d) $11.07 million
(e) 1 + 1 < 11 but 9.2 + 9.2 > 11.07.
8.6. (a) $3.92 million, (b) $8.77 million, (c) $10.40 million.
8.7. (b) becomes $9.96 million and (c) becomes $11.82 million.
8.8. Marginal VaR is the rate of change of VaR with the amount invested
in the ith asset. Incremental VaR is the incremental effect of the ith asset
on VaR (i.e., the difference between VaR with and without the asset).
Component VaR is the part of VaR that can be attributed the ith asset
(the sum of component VaRs equals the total VaR).
8.9. The probability of 17 or more exceptions is
1 - BINOMDIST(16,1000,0.01,TRUE)
or 2.64%. The model should be rejected at the 5% confidence level.
8.10. Bunching is the tendency for exceptions to be bunched rather than
occurring randomly throughout the time period considered.
8.11. Either historical data or brainstorming by senior management can
be used to develop extreme scenarios.
8.12. We are interested in the standard deviation of R1 + R2 + ... + Rn
where Ri is the return on day i. This is
where
is the standard deviation of Ri and
is the correlation between Ri and
Rj. In this case,
for all i, and
when i > j. After further
algebraic manipulations this leads directly to equation (8.3).
8.13. The probability of 5 or more exceptions is
1 - BINOMDIST(4,250,0.01,TRUE)
or 10.8%.
CHAPTER 9
9.1. The assumption is that the statistical process driving changes in
market variables over the next day is the same as that over the last 500 days.
9.2.
This shows that, as
approaches 1, the weights approach \/n.
9.3. The tenth-worst outcome is a return of —3.78%. The estimate of the
1-day 99% VaR is therefore $3.78 million.
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9.4. The 1-day 99% VaR is estimated as $1.36 million. This is much less
than that given in Problem 9.3 because most of the really bad returns were
more than 500 days ago and carry relatively little weight.
9.5. The volatility is initially 1.14% per day. It varies from 0.428% per
day to 2.97% per day. After adjustment, the tenth-worst outcome is
1.77%. The VaR estimate is therefore $1.77 million.
9.6. The VaR estimates given by u equal to 0.005, 0.01, and 0.015 are
3.34, 3.34, and 3.30, respectively.
9.7. The standard error of the VaR estimate is
or $0.69 million.
CHAPTER 10
10.1. The standard deviation of the daily change in the investment in
each asset is $1,000. The variance of the portfolio's daily change is
1,0002 + 1,0002 + 2 x 0.3 x 1,000 x 1,000 = 2,600,000
The standard deviation of the portfolio's daily change is the square root
of this, or $1,612.45. The 5-day 99% VaR is therefore
2.33 x
x 1,612.45 = $8,401
10.2. The three alternative procedures mentioned in the chapter for
handling interest rates when the model-building approach is used to
calculate VaR involve (a) the use of the duration model, (b) the use of
cash-flow mapping, and (c) the use of principal components analysis.
10.3. When a final exchange of principal is added in, the floating side is
equivalent to a zero-coupon bond with a maturity date equal to the date
of the next payment. The fixed side is a coupon-bearing bond, which is
equivalent to a portfolio of zero-coupon bonds. The swap can therefore
be mapped into a portfolio of zero-coupon bonds with maturity dates
corresponding to the payment dates. Each of the zero-coupon bonds can
then be mapped into positions in the adjacent standard-maturity zerocoupon bonds.
10.4.
= 56 x
The
standard deviation of
is 56 x 1.5 x
0.007 = 0.588. It follows that the 10-day 99% VaR for the portfolio is
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Answers to Problems and Questions
10.6. The 6. 5-year cash flow is equivalent to a position of $48.56 in a
5-year zero-coupon bond and a position of $605.49 in a 7-year zerocoupon bond. The equivalent 5-year and 7-year cash flows are
48.56 x 1.065 = 64.98 and 605.49 x 1.077 = 972.28.
10.7. A similar calculation to that in the text shows that $37,397 of the
value is allocated to the 3-month bond worth and $11,793 of the value is
allocated to the 6-month bond.
10.8. The daily variance of the portfolio is
62 x 202 + 42 x 82 = 15,424
and the daily standard deviation is
N(-1.282) = 0.9, the 5-day 90% VaR is
124.19 x
= $124.19.
Since
x 1.282 = $356.01
10.9. (a) 3.26, (b) 63.87.
10.10. The delta of the options is the rate of change of the value of the
options with respect to the price of the asset. When the asset price
increases by a small amount, the value of the options decreases by 30
times this amount. The gamma of the options is the rate of change of
their delta with respect to the price of the asset. When the asset price
increases by a small amount, the delta of the portfolio decreases by five
times this amount.
In this case,
= -0.10,
= 36.03, and
= -32.415.
The mean change in the portfolio value in 1 day is —0.1 and the standard
deviation of the change in 1 day is
= 6.002. The Skewness is
Using only the first two moments, we find that the 1-day 99% VaR is
$14.08. When three moments are considered in conjunction with a
Cornish-Fisher expansion, it is $14.53.
10.11. Define as the volatility per year,
as the change in in 1 day,
and
as the proportional change in in 1 day. We measure in as a
multiple of 1% so that the current value of is 1 x
= 15.87. The
delta-gamma-vega model is
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or
where
which simplifies to
The change in the portfolio value now depends on two market variables.
Once the daily volatility of and the correlation between a nd S have
been estimated, we can estimate moments of
and use a CornishFisher expansion.
10.12. The change in the value of an option is not linearly related to the
change in the value of the underlying variables. When the change in the
values of underlying variables is normal, the change in the value of the
option is not normal. The linear model assumes that it is normal and is,
therefore, only an approximation.
10.13. The contract is a long position in a sterling bond combined with a
short position in a dollar bond. The value of the sterling bond is
or $1.492 million. The value of the dollar bond is
or $1.463 million. The variance of the change in the value
of the contract in 1 day is
1.4922 x 0.00062 + 1.4632 x 0.00052
- 2 x 0.8 x 1.492 x 0.0006 x 1.463 x 0.0005 = 0.000000288
The standard deviation is therefore $0.000537 million. The 10-day 99%
VaR is 0.000537 x
x 2.33 = $0.00396 million.
CHAPTER 11
11.1 Moody's has 19 ratings (excluding the "in default" rating): Aaa,
Aal, Aa2, Aa3, Al, A2, A3, Baal, Baa2, Baa3, Bal, Ba2, Ba3, Bl, B2,
B3, Caal, Caa2, Caa3.
11.2. S&P has 19 ratings (excluding the "in default" rating): AAA, AA+,
AA, A A - , A+, A, A - , BBB+, BBB, BBB-, BB+, BB, B - , B+, B, B - ,
CCC+, CCC, C C C - .
11.3. Average default intensity is
year.
where
= 0.9379. It is 6.41% per
11.4. Conditional on no default by year 2, the probability of no default
by year 3 is 0.8926/0.9526 = 0.9370. Average default intensity for the third
year is
where
= 0.9370. It is 6.51% per year.
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Answers to Problems and Questions
11.5. From equation (11.3) the average default intensity over the 3 years
is 0.0050/(1 - 0.3) = 0.0071, or 0.71% per year.
11.6. From equation (11.3) the average default intensity over 5 years is
0.0080/(1-0.4), or 1.333% per year. Similarly the average default
intensity over 3 years is 1.1667% per year. This means that the average
default intensity for years 4 and 5 is 1.58%.
11.7. Real-world probabilities of default should be used for calculating
credit value at risk. Risk-neutral probabilities of default should be used
for adjusting the price of a derivative for default.
11.8. The recovery rate for a bond is the value of the bond immediately
after the issuer defaults as a percentage of its face value.
11.9. The first number in the second column of Table 11.4 is calculated as
or 0.04% per year. Other numbers in the column are calculated similarly.
The numbers in the fourth column of Table 11.5 are the numbers in the
second column of Table 11.4 multiplied by one minus the expected
recovery rate. In this case, the expected recovery rate is 0.4.
11.10. The bond's market value is 96.19. Its risk-free value is 103.66. If Q
is the default probability per year, the loss from defaults is 272.69Q. The
implied probability of default is therefore 2.74% per year.
11.11. The market price of the first bond is 98.35 and its risk-free value is
101.23. If Q1 is the default probability at times 0.5, 1.5, and 2.5 years the
loss from defaults for the first bond is 178.31Q1. It follows that
Q1 =0.0161. If Q2 is the probability of default at times 3.5 and 4.5,
the loss from default from the second bond is 180.56Q1 + 108.53Q2. The
market price for the second bond is 96.24 and its risk-free value is 101.97.
It follows that 180.56Q1 + 108.5302 = 5.73 and Q2 = 0.0260.
11.12. We can assume that the principal is paid and received at the end of
the life of the swap without changing the swap's value. If the spread were
zero, then the present value of the floating payments per dollar of principal
would be 1. The payment of LIBOR plus the spread therefore has a present
value of 1 + V. The payment of the bond cash flows has a present value per
dollar of principal of B*. The initial payment required from the payer of
the bond cash flows per dollar of principal is 1 — B. (This may be negative;
an initial amount of B — 1 is then paid by the payer of the floating rate.)
Because the asset swap is initially worth zero, we have
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so that
V = B* -B
11.13. The value of the debt in Merton's model is V0 — E0, or
If the credit spread is s, this should equal
Substituting
so that
we get
so that
11.14. In this case, E0 = 2,
= 0.50, D = 5, r = 0.04, and T = 1 .
Solving the simultaneous equations gives V0 = 6.80 and
= 14.82.
The probability of default is N(-d2), or 1.15%.
11.15. Suppose that the principal is $100. The asset swap is structured so
that the $10 is paid initially. After that, $2.50 is paid every 6 months. In
return, LIBOR plus a spread is received on the principal of $100. The
present value of the fixed payments is
The spread over LIBOR must therefore have a present value of 5.3579.
The present value of $1 received every 6 months for 5 years is 8.5105. The
spread received every 6 months must therefore be
5.3579/8.5105 = $0.6296
The asset swap spread is therefore 2 x 0.6296 = 1.2592% per annum.
CHAPTER 12
12.1. The new transaction will increase the bank's exposure to the
counterparty if it tends to have a positive value whenever the existing
contract has a positive value and a negative value whenever the existing
contract has a negative value. However, if the new transaction tends to
offset the existing transaction, it is likely to have the incremental effect of
reducing credit risk.
12.2. Equation (12.3) gives the relationship between
and
This
involves QA(T) and QB(T). These change as we move from the real world
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Answers to Problems and Questions
to the risk-neutral world. It follows that the relationship between
and
in the real world is not the same as in the risk-neutral world. If
is the same in the two worlds, then
is not, and vice versa.
12.3. When securities are pledged as collateral, the haircut is the discount
applied to their market value for margin calculations. A company's own
equity would not be good collateral. When the company defaults on its
contracts, its equity is likely to be worth very little.
12.4. In Vasicek's model and Credit Risk Plus, a credit loss recognized
when a default occurs. In CreditMetrics, both downgrades and defaults
lead to credit losses. In Vasicek's model, a Gaussian copula model of time
to default is used. In Credit Risk Plus, a probability distribution is
assumed for the default rate per year. In CreditMetrics, a Gaussian
copula model is used to define rating transitions.
12.5. The binomial correlation measure is 0.156.
12.6. The statements in (a) and (b) are true. The statement in (c) is not.
Suppose that
and
are the exposures to X and Y. The expected value
of
is the expected value of
plus the expected value of
The
same is not true of 95% confidence limits.
12.7. The cost of defaults is
where u is percentage loss from defaults
during the life of the contract and v is the value of an option that pays off
max(150ST - 100, 0), where ST is the AUD/USD exchange rate in 1 year
(USD per AUD). The value of u is
The variable v is 150 times the value of a call option to buy 1 AUD for
0.6667. This is 4.545. It follows that the cost of defaults is
4.545 x 0.009950, or 0.04522.
12.8. In this case, the cost of defaults is
where
is the value of a 6-month call option on 150 AUD with a strike price of
$100, and
is the value of a similar 1-year option.
= 3.300 and
= 4.545. The cost of defaults is 0.04211.
12.9. Assume that defaults happen only at the end of the life of the
forward contract. In a default-free world, the forward contract is the
combination of a long European call and a short European put where the
strike price of the options equals the delivery price and the maturity of the
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447
options equals the maturity of the forward contract. If the no-default
value of the contract is positive at maturity, the call has a positive value
and the put is worth zero. The impact of defaults on the forward contract
is the same as that on the call. If the no-default value of the contract is
negative at maturity, the call has a zero value and the put has a positive
value. In this case, defaults have no effect. Again the impact of defaults
on the forward contract is the same as that on the call. It follows that the
contract has a value equal to a long position in a call that is subject to
default risk and short position in a default-free put.
12.10. Suppose that the forward contract provides a payoff at time T.
With our usual notation, the value of a long forward contract is
ST
—
(see Appendix A). The credit exposure on a long forward
contract is therefore max(ST —
, 0); that is, it is a call on the asset
price with strike price
Similarly, the credit exposure on a short
forward contract is max(
— ST, 0); that is, it is a put on the asset
price with strike price
The total credit exposure is therefore a
straddle with strike price
12.11. As time passes, there is a tendency for the currency which has the
lower interest rate to strengthen. This means that a swap where we are
receiving this currency will tend to move in the money (i.e., have a
positive value). Similarly, a swap where we are paying the currency will
tend to move out of the money (i.e., have a negative value). From this it
follows that our expected exposure on the swap where we are receiving
the low-interest currency is much greater than our expected exposure on
the swap where we are receiving the high-interest currency. We should
therefore look for counterparties with a low credit risk on the side of the
swap where we are receiving the low-interest currency. On the other side
of the swap, we are far less concerned about the creditworthiness of the
counterparty.
CHAPTER 13
13.1. Both provide insurance against a particular company defaulting
during a period of time. In a credit default swap, the payoff is the
notional principal amount multiplied by one minus the recovery rate.
In a binary swap the payoff is the notional principal.
13.2. The seller receives 300,000,000 x 0.0060 x 0.5 = $900,000 at times
0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0 years. The seller also receives a final
accrual payment of about $300,000 (= $300,000,000 x 0.060 x 2/12) at
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the time of the default (4 years and 2 months). The seller pays
300,000,000 x 0.6 = $180,000,000 at the time of the default.
13.3. Sometimes there is physical settlement and sometimes there is cash
settlement. In the event of a default when there is physical settlement, the
buyer of protection sells bonds issued by the reference entity for their face
value. Bonds with a total face value equal to the notional principal can be
sold. In the event of a default when there is cash settlement, a calculation
agent estimates the value of the Cheapest-to-deliver bond issued by the
reference entity a specified number of days after the default event. The
cash payoff is then based on the excess of the face value of these bonds
over the estimated value.
13.4. A cash CDO is created from a bond portfolio. The returns from the
bond portfolio flow to a number of tranches (i.e., different categories of
investors). The tranches differ as far as the credit risk they assume. The
first tranche might have an investment in 5% of the bond portfolio and be
responsible for the first 5% of losses. The next tranche might have an
investment in 10% of the portfolio and be responsible for the next 10%
of the losses, and so on. In a synthetic CDO there is no bond portfolio.
Instead, a portfolio of credit default swaps is sold and the resulting credit
risks are allocated to tranches in a similar way to that just described.
13.5. In a first-to-default basket CDS, there are a number of reference
entities. When the first one defaults, there is a payoff (calculated in the
usual way for a CDS) and the basket CDS terminates. The value of the
protection given by the first-to-default basket CDS decreases as the
correlation between the reference entities in the basket increases. This is
because the probability of a default decreases as the correlation increases.
In the limit when the correlation is 1, there is in effect only one company
and the probability of a default is quite low.
13.6. Risk-neutral default probabilities are backed out from credit default
swaps or bond prices. Real-world default probabilities are calculated from
historical data.
13.7. Suppose a company wants to buy some assets. If a total return swap
is used, a financial institution buys the assets and enters into a swap with
the company where it pays the company the return on the assets and
receives from the company LIBOR plus a spread. The financial institution has less risk than it would have if it lent the company money and
used the assets as collateral. This is because, in the event of a default by
the company, it owns the assets.
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13.8. An analysis similar to that in Tables 13.1 to 13.4 gives the PV of
expected payments as 3.7364s, the PV of the expected payoff as 0.0838,
and the PV of the expected accrual payment as 0.05985. The credit default
swap spread is 221 basis points.
13.9. If the credit default swap spread is 150 basis points, the value of the
swap to the buyer of protection is
0.0838 - (3.7364 + 0.0598) x 0.0150 = 0.0269
per dollar of notional principal.
13.10. If the swap is a binary CDS, the present value of expected payoffs
is 0.1197 and the credit default swap spread is 315 basis points.
13.11. A 5-year nth-to-default credit default swap works in the same way
as a regular credit default swap except that there is a basket of companies.
The payoff occurs when the nth default from the companies in the basket
occurs. After the nth default has occurred, the swap ceases to exist. When
n = 1 (so that the swap is a "first to default"), an increase in the default
correlation lowers the value of the swap to the protection buyer. When
n = 25 (so that the swap is a 25th to default), an increase in the default
correlation increases the value of the swap to the protection buyer.
13.12. The recovery rate of a bond is usually defined as the value of the
bond a few days after a default occurs as a percentage of the bond's face
value.
13.13. The payoff from a plain vanilla CDS is 1 — R times the payoff
from a binary CDS with the same principal. The payoff always occurs at
the same time on the two instruments. It follows that the regular payments on a new plain vanilla CDS must be 1 - R times the payments on a
new binary CDS. Otherwise there would be an arbitrage opportunity.
13.14. In the case of a total return swap, a company receives (pays) the
increase (decrease) in the value of the bond. In a regular swap this does
not happen.
13.15. When a company enters into a long (short) forward contract it is
obligated to buy (sell) the protection given by a specified credit default
swap with a specified spread at a specified future time. When a company
buys a call (put) option contract, it has the option to buy (sell) the
protection given by a specified credit default swap with a specified spread
at a specified future time. Both contracts are normally structured so that
they cease to exist if a default occurs during the life of the contract.
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13.16. A credit default swap insures a corporate bond issued by the
reference entity against default. Its approximate effect is to convert the
corporate bond into a risk-free bond. The buyer of a credit default swap
has therefore chosen to exchange a corporate bond for a risk-free bond
This means that the buyer is long a risk-free bond and short a similar
corporate bond.
13.17. Payoffs from credit default swaps depend on whether a particular
company defaults. Arguably some market participants have more
information about this that others (see Business Snapshot 13.2).
13.18. Real-world default probabilities are less than risk-neutral default
probabilities. It follows that the use of real-world default probabilities
will tend to understate the value of the protection.
13.19. In an asset swap the bond's promised payments are swapped for
LIBOR plus a spread. In a total return swap the bond's actual payments
are swapped for LIBOR plus a spread.
13.20. Using equation (13.2), we find that the probability of default
conditional on a factor value of M is
For M equal to - 2 , — 1, 0, 1, 2, the probabilities of default are 0.135,
0.054, 0.018, 0.005, 0.001, respectively. To six decimal places, the probability of more that 10 defaults for these values of M can be calculated
using the BINOMDIST function in Excel. They are 0.959284, 0.79851,
0.000016, 0, 0, respectively.
13.21. For a CDO squared we form a portfolio of CDO tranches and
tranche the default losses in a similar way to Figure 13.3. For a CDO
cubed we form a portfolio of CDO squared tranches and tranche the
default losses in a similar way to Figure 13.3.
CHAPTER 14
14.1. The definition includes all internal risks and external risks except
reputational risk and risks resulting from strategic decisions.
14.2. Based on the results reported in Section 14.4, the loss would be
100 x 3 0 . 2 3 , or $127.8 million.
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14.3.
,
When x = 20, the probability is 0.1. This
means that K = 1.0986. The probability of the specified losses being
exceeded are (a) 5.74%, (b) 3.30%, and (c) 1.58%.
14.4. Moral hazard is handled by deductibles and by making premiums
dependent on past claims. Adverse selection is handled by finding out as
much as possible about a driver before insurance is granted and then
modifying premiums as more information on the driver becomes available.
14.5. CEOs must prepare a statement asserting that the financial statements are accurate. They must return bonuses in the event that there is a
restatement of financial statements.
14.6. If a trader operates within established risk limits and takes a loss, it
is part of market risk. If risk limits are violated, the loss becomes
classified as an operational risk.
14.7. (a) It is unlikely that an individual would not look after his or her
health because of the existence of a life insurance contract. But it has been
known for the beneficiary of a life insurance contract to commit murder
to receive the payoff from the contract! (b) Individuals with short life
expectancies are more likely to buy life insurance than individuals with
long life expectancies.
14.8. External loss data is data relating to the losses of other banks. It is
data obtained from sharing agreements with other banks or from data
vendors. External data is used to determine relative loss severity. It can be
a useful indicator of the ratio of mean losses in Business Unit A to mean
losses in Business Unit B or the ratio of the standard deviation of losses
in Business Unit A to the standard deviation of losses in Business Unit B.
14.9. The Poisson distribution is commonly used for loss frequency. The
lognormal distribution is commonly used for loss severity.
14.10. Two examples of key risk indicators are staff turnover and number
of failed transactions.
14.11. When the loss frequency is 3, the mean total loss is about 3.3 and
the standard deviation is about 2.0. When the loss frequency is increased
to 4, the mean loss is about 4.4 and the standard deviation is about 2.4.
CHAPTER 15
15.1. Leverage and Crashophobia.
15.2. Uncertain volatility and jumps.
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15.3. When plain vanilla call and put options are being priced, traders do
use the Black-Scholes model as an interpolation tool. They calculate
implied volatilities for the options that are actively traded. By interpolating between strike prices and between times to maturity, they estimate
implied volatilities for other options. These implied volatilities are then
substituted into Black-Scholes to calculate prices for these options.
Black-Scholes is more than an interpolation tool when used for hedging.
15.4. 13.45%. We get the same answer by (a) interpolating between strike
prices of 1.00 and 1.05 and then between maturities of 6 months and
1 year and (b) interpolating between maturities of 6 months and 1 year
and then between strike prices of 1.00 and 1.05.
15.5. The models of physics describe the behavior of physical processes.
The models of finance ultimately describe the behavior of human beings.
15.6. It might notice that it is getting a large amount of business of a
certain type because it is quoting prices different from its competitors.
The pricing differences may also become apparent if it decides to unwind
transactions and approaches competitors for quotes. Also, it might
subscribe to a service where it obtains the average price quotes by dealers
for particular transactions.
15.7. A loss equal to half the bid-offer spread is recognized when
positions are liquidated. Liquidity VaR takes this loss into account.
15.8. Liquidity black holes occur when most market participants want to
be on one side of a market. Regulation is liable to lead to liquidity black
holes. This is because when all financial institutions are regulated in the
same way they tend to want to respond to external economic events in the
same way.
15.9. Within-model hedging involves hedging against changes in variables
that the model assumes to be stochastic. Outside-model hedging involves
hedging against parameters that the model assumes to be constant.
15.10. The Black-Scholes model assumes that the probability distribution of the stock price in 1 month is lognormal. In this case it is clearly
not lognormal. Possibly it consists of two lognormal distributions superimposed upon each other and is bimodal. Black-Scholes is clearly
inappropriate.
15.11. In this case the probability distribution of the exchange rate has a
thin left tail and a thin right tail relative to the lognormal distribution.
Deep-out-of-the-money calls and puts will have relatively low prices.
Answers to Problems and Questions
453
15.12. The term "marking to market" refers to the practice of revaluing
instruments (usually daily) so that they are consistent with the market.
The prices calculated for actively traded products do reflect market prices.
The model is used merely as an interpolation tool. The term "marking to
market" is therefore accurate for these products. The prices for structured
products depend on the model being used. Hence the term "marking to
model".
15.13. Hedge funds are not regulated in the same way as other financial
institutions and can therefore be contrarian investors buying whenever
everyone else is selling and selling whenever everyone else is buying.
However, black holes can be created when large numbers of hedge funds
follow similar trading strategies.
CHAPTER 16
16.1. Economic capital is a bank's own estimate of the capital it requires.
Regulatory capital is the capital it is required to keep by bank supervisors.
16.2. A company with an AA rating has a 0.03% chance of defaulting in
1 year.
16.3. Business risk includes risks relating to strategic decisions and
reputation.
16.4. The models used for economic capital are likely to be broadly
similar to those used to calculate regulatory capital in the case of market
risk and operational risk. When calculating credit risk economic capital, a
bank may consider it appropriate to use a different credit correlation
model and different correlation parameters from those used in regulatory
capital calculations.
16.5. The 99.97% worst-case value of the logarithm of the loss is
0.5 + 4 x 3.43 = 14.23. The 99.97% worst-case loss is therefore $1,510
million. From the properties of the lognormal distribution, the expected
loss is exp(0.5+ 4 2 /2), or $4,915. The capital requirement is therefore
$1.505 million.
16.6. The economic capital for Business Unit 1 is 96.85. The economic
capital for Business Unit 2 is 63.87. The total capital is 124.66.
16.7. The incremental effect of Business Unit 1 on total economic
capital is 60.78. The incremental effect of Business Unit 2 on total
economic capital is 27.81. This suggests that 60.78/(60.78 + 27.81), or
68.61%, of economic capital should be allocated to Business Unit 1 and
27.81/(60.78 + 27.81), or 31.39%, to Business Unit 2. The marginal
454
Answers to Problems and Questions
effect of increasing the size of Business Unit 1 by 0.5% is 0.4182. The
marginal effect of increasing the size of Business Unit 2 by 0.5% is
0.2056. Euler's theorem is satisfied because the total economic capital is
approximately equal to the sum of 0.4182/0.005 and 0.2056/0.005.
16.8. The capital is 40 — 2 = $38 million and the return before tax is
12 - 5 - 2 = $5 million. The before-tax RAROC is therefore 13.2%. In
practice, the allocation of diversification benefits to this venture might
reduce capital and increase RAROC.
16.9. RAROC can be used to compare the past performance of different
business units or to project the expected future performance of business
units.
CHAPTER 17
17.1. A day's HDD is max(0, 65 - A) and a day's CDD is max(0, A - 65),
where A is the average of the highest and lowest temperature during the day
at a specified weather station, measured in degrees Fahrenheit.
17.2. It is an agreement by one side to deliver a specified amount of gas at
a roughly uniform rate during a month to a particular hub for a specified
price.
17.3. The average temperature each day is 75° Fahrenheit. The CDD each
day is therefore 10 and the cumulative CDD for the month is
10 x 31 = 310. The payoff from the call option is therefore
(310 - 250) x 5,000 = $300,000
17.4. Unlike most commodities electricity cannot be stored easily. If the
demand for electricity exceeds the supply, as it sometimes does during the
air-conditioning season, the price of electricity in a deregulated environment will skyrocket. When supply and demand become matched again,
the price will return to former levels.
17.5. HDD is max(65 — A, 0), where A is the average of the maximum
and minimum temperature during the day. This is the payoff from a put
option on A with a strike price of 65. CDD is max(A — 65, 0). This is the
payoff from call option on A with a strike price of 65.
17.6. It would be useful to calculate the cumulative CDD for July of each
year of the last 50 years. A linear regression relationship
CDD = a + bt + e
Answers to Problems and Questions
455
could then be estimated, where a and b are constants, t is the time in years
measured from the start of the 50 years, and e is the error. This relationship allows for linear trends in temperature through time. The expected
CDD for next year (year 51) is then a + 5\b. This could be used as an
estimate of the forward CDD.
17.7. The volatility of the one-year forward price will be less than the
volatility of the spot price. This is because, when the spot price changes
by a certain amount, mean reversion will cause the forward price will
change by a lesser amount.
17.8. The energy producer faces quantity risks and price risks. It can use
weather derivatives to hedge the quantity risks and energy derivatives to
hedge against the price risks.
17.9. A 5 x 8 contract for May 2006 is a contract to provide electricity
for 5 days per week during the off-peak period (11 p.m. to 7 a.m.). When
daily exercise is specified, the holder of the option is able to choose each
weekday whether he or she will buy electricity at the strike price at the
agreed rate. When there is monthly exercise, he or she chooses once at the
beginning of the month whether electricity is to be bought at the strike
price at the agreed rate for the whole month. The option with daily
exercise is worth more.
17.10. CAT bonds (catastrophe bonds) are an alternative to reinsurance
for an insurance company that has taken on a certain catastrophic risk
(e.g., the risk of a hurricane or an earthquake) and wants to get rid of it.
CAT bonds are issued by the insurance company. They provide a higher
rate of interest than government bonds. However, the bondholders agree
to forego interest, and possibly principal, to meet any claims against the
insurance company that are within a prespecified range.
17.11. The CAT bond has very little systematic risk. Whether a particular
type of catastrophe occurs is independent of the return on the market.
The risks in the CAT bond are likely to be largely "diversified away" by
the other investments in the portfolio. A B-rated bond does have systematic risk that cannot be diversified away. It is likely therefore that the
CAT bond is a better addition to the portfolio.
17.12. It means that the price of the energy source will be pulled back to
a long-run average level. Electricity has the highest mean-reversion rate;
oil has the lowest.
Glossary of Terms
Accrued Interest
payment date.
The interest earned on a bond since the last coupon
Add-on Factor When the credit equivalent amount for a derivatives
transaction is being calculated, this is the percentage of principal added
to the current exposure to allow for possible future changes in the value
of the derivative.
Advanced Measurement Approach The way in which the most sophisticated banks will be allowed to calculate regulatory capital for operational
risk under Basel II.
Adverse Selection The phenomenon that, if an insurance company
offers the same premiums to everyone, it tends to end up providing
coverage for the worst risks.
American Option
life.
Analytic Result
An option that can be exercised at any time during its
Result where answer is in the form of an equation.
Arbitrage A trading strategy that takes advantage of two or more securities being mispriced relative to each other.
Arbitrage Pricing Theory A theory where the return from an investment
is assumed to depend on several factors.
Arbitrageur
An individual engaging in arbitrage.
458
Glossary of Terms
Asian Option An option with a payoff dependent on the average price
of the underlying asset during a specified period.
Ask Price The price that a dealer is offering to sell an asset. (Also called
the offer price.)
Asked Price
See Ask Price.
Asset Swap
a spread.
Exchanges the promised coupon on a bond for LIBOR plus
At-the-money Option An option in which the strike price equals the
price of the underlying asset.
Autocorrelation The correlation between the value of a variable and
the value of the same variable k days later (where k is referred to as the
time lag).
Average Price Call Option An option giving a payoff equal to the
greater of zero and the amount by which the average price of the asset
exceeds the strike price.
Average Price Put Option An option giving a payoff equal to the greater
of zero and the amount by which the strike price exceeds the average price
of the asset.
Back Testing
data.
Testing a value-at-risk or other model using historical
Backwards Induction A procedure for working from the end of a tree to
its beginning in order to value an option.
Bankruptcy Costs Costs such as lost sales, loss of key managers, and
professional fees arising from a declaration of bankruptcy. These costs
are not associated with the adverse events leading to bankruptcy.
Barrier Option An option whose payoff depends on whether the path of
the underlying asset has reached a barrier (i.e., a certain predetermined
level).
Basel I The first international agreement on the regulation of banks
in 1988.
Basel II New international regulations for calculating bank capital
expected to come into effect in 2007.
Basic Indicator Approach The simplest way of calculating regulatory
capital for operational risk under Basel II.
Basis The difference between the spot price and the futures price of a
commodity.
Glossary of Terms
459
Basis Point When used to describe an interest rate, a basis point is one
hundredth of one percent (= 0.01%).
Basis Risk The risk to a hedger arising from uncertainty about the basis
at a future time.
Basket Credit Default Swap Credit default swap where there are several
reference entities.
Basket Option Option on a portfolio of assets.
Beta A measure of the systematic risk of an asset.
Bid-Ask Spread See Bid-Offer Spread.
Bid-Offer Spread The amount by which the offer (or ask) price exceeds
the bid price.
Bid Price The price that a dealer is prepared to pay for an asset.
Binary Credit Default Swap Instrument where there is a fixed dollar
payoff in the event of a default by a particular company.
Binary Option Option with a discontinuous payoff, for example, a cashor-nothing option or an asset-or-nothing option.
Binomial Model A model where the price of an asset is monitored over
successive short periods of time. In each short period, it is assumed that
only two price movements are possible.
Binomial Tree A tree that represents how an asset price can evolve
under the binomial model.
Bivariate Normal Distribution A distribution for two correlated variables, each of which is normal.
Black's Model An extension of the Black-Scholes model for valuing
European options on futures contracts. It is used extensively in practice to
value European options when the distribution of the asset price at
maturity is assumed to be lognormal.
Black-Scholes Model A model for pricing European options on stocks,
developed by Fischer Black, Myron Scholes, and Robert Merton.
Bond Option An option where a bond is the underlying asset.
Bond Yield Discount rate which, when applied to all the cash flows of a
bond, causes the present value of the cash flows to equal the bond's
market price.
Bootstrap Method A procedure for calculating the zero-coupon yield
curve from market data. Also a statistical procedure for calculating
confidence levels when distributions are determined empirically.
460
Glossary of Terms
Bunching A tendency for days when the loss is greater than the value at
risk to be bunched close together.
Business Risk When used for a bank, this refers to strategic risk (related
to a bank's decision to enter new markets and develop new products) and
reputation risk.
Calendar Days Includes every day.
Calibration Method for implying a model's parameters from the prices
of actively traded options.
Callable bond A bond containing provisions that allow the issuer to
buy it back at a predetermined price at certain times during its life.
Call Option An option to buy an asset at a certain price by a certain
date.
Cancelable Swap Swap that can be canceled by one side on prespecified
dates.
Cap See Interest Rate Cap.
Capital Asset Pricing Model A model relating the expected return on an
asset to its beta.
Caplet One component of an interest rate cap.
Cap Rate The rate determining payoffs in an interest rate cap.
Cash Flow Mapping A procedure for representing an instrument as a
portfolio of zero-coupon bonds for the purpose of calculating value at
risk.
Cash Settlement Procedure for settling a contract in cash rather than by
delivering the underlying asset.
CAT Bond Bond where the interest and, possibly, the principal paid are
reduced if a particular category of "catastrophic" insurance claims exceed
a certain amount.
CDD Cooling degree days. The maximum of zero and the amount by
which the daily average temperature is greater than 65° Fahrenheit. The
average temperature is the average of the highest and lowest temperatures
(midnight to midnight).
CDO See Collateralized Debt Obligation.
CDO Squared An instrument in which the default risks in a portfolio of
CDO tranches are allocated to new securities.
CDX An index of the credit quality of 125 North American investmentgrade companies.
Glossary of Terms
461
Cholesky Decomposition
normal distribution.
Method of sampling from a multivariate
Clean Price of Bond The quoted price of a bond. The cash price paid
for the bond (or dirty price) is calculated by adding the accrued interest to
the clean price.
Clearinghouse A firm that guarantees the performance of the parties in
an exchange-traded derivatives transaction. (Also referred to as a clearing
corporation.)
Clearing margin
A margin posted by a member of a clearinghouse.
Coherent Risk Measure
conditions.
Collar
A risk measure that satisfies a number of
See Interest Rate Collar.
Collateralization A system for posting collateral by one or both parties
in a derivatives transaction.
Collateralized Debt Obligation A way of packaging credit risk. Several
classes of securities (known as tranches) are created from a portfolio of
bonds and there are rules for determining how the cost of defaults are
allocated to classes.
Component VaR VaR corresponding to a component of a portfolio.
Defined so that the sum of the component VaRs for the components of a
portfolio equals the VaR for the whole portfolio.
Compounding Frequency
Compound Option
Compounding Swap
paid.
This defines how an interest rate is measured.
An option on an option.
Swap where interest compounds instead of being
Conditional Value at Risk (C-VaR)
See Expected Shortfall.
Confirmation Contract confirming verbal agreement between two parties
to a trade in the over-the-counter market.
Consumption Asset
ment.
An asset held for consumption rather than invest-
Continuous Compounding A way of quoting interest rates. It is the limit
as the assumed compounding interval is made smaller and smaller.
Convenience Yield A measure of the benefits from ownership of an
asset that are not obtained by the holder of a long futures contract on the
asset.
462
Glossary of Terms
Conversion Factor Factor multiplied by principal to convert an offbalance-sheet item to its credit equivalent amount.
Convertible Bond A corporate bond that can be converted into a
predetermined amount of the company's equity at certain times during
its life.
Convexity A measure of the curvature in the relationship between bond
prices and bond yields.
Convexity Adjustment An overworked term. For example, it can refer to
the adjustment necessary to convert a futures interest rate to a forward
interest rate. It can also refer to the adjustment to a forward rate that is
sometimes necessary when instruments are valued.
Cooke Ratio
Ratio of capital to risk-weighted assets under Basel I.
Copula A way of defining the correlation between variables with known
distributions.
Cornish-Fisher Expansion An approximate relationship between the
fractiles of a probability distribution and its moments.
Cost of Carry The storage costs plus the cost of financing an asset minus
the income earned on the asset.
Counterparty
Coupon
The opposite side in a financial transaction.
Interest payment made on a bond.
Covariance Measure of the linear relationship between two variables
(equals the correlation between the variables times the product of their
standard deviations).
Covered Call A short position in a call option on an asset combined
with a long position in the asset.
Crashophobia The fear of a stock market crash similar to that in 1987
that some people claim causes market participants to increase the value of
deep-out-of-the-money put options.
Credit Default Swap An instrument that gives the holder the right to
sell a bond for its face value in the event of a default by the issuer.
Credit Derivative A derivative whose payoff depends on the creditworthiness of one or more companies or countries.
Credit Equivalent Amount Size of loan that is considered equivalent to
an off-balance-sheet transaction in Basel I.
Credit Rating
A measure of the creditworthiness of a bond issue.
Glossary of Terms
463
Credit Ratings Transition Matrix A table showing the probability that a
company will move from one credit rating to another during a certain
period of time.
Credit Risk The risk that a loss will be experienced because of a default
by the counterparty in a derivatives transaction.
Credit Risk Migration Movement of a company from one rating category
to another.
Credit Risk Plus
A procedure for calculating credit value at risk.
Credit Value at Risk The credit loss that will not be exceeded at some
specified confidence level.
CreditMetrics A procedure for calculating credit value at risk.
Cumulative Distribution Function The probability that a variable will
be less than x as a function of x.
Currency Swap A swap where interest and principal in one currency are
exchanged for interest and principal in another currency.
Day Count
A convention for quoting interest rates.
Day Trade
A trade that is entered into and closed out on the same day.
Default Correlation Measures the tendency of two companies to default
at about the same time.
Default Intensity See Hazard Rate.
Delivery Price Price that will be paid or received in a forward contract.
Delta The rate of change of the price of a derivative with the price of the
underlying asset.
Delta Hedging A hedging scheme that is designed to make the price of a
portfolio of derivatives insensitive to small changes in the price of the
underlying asset.
Delta-neutral Portfolio A portfolio with a delta of zero so that there is
no sensitivity to small changes in the price of the underlying asset.
DerivaGem Software for valuing options, available on the author's
website.
Derivative An instrument whose price depends on, or is derived from,
the price of another asset.
Deterministic Variable A variable whose future value is known.
Dirty Price of Bond Cash price of bond.
Discount Bond See Zero-coupon Bond.
464
Discount Instrument
provides no coupons.
Glossary of Terms
An instrument, such as a Treasury bill, that
Discount Rate The annualized dollar return on a Treasury bill or
similar instrument expressed as a percentage of the final face value.
Distance to Default The number of standard deviations that the value
of a company's assets must move for a default to be triggered.
Dividend A cash payment made to the owner of a stock.
Dividend Yield The dividend as a percentage of the stock price.
Down-and-in Option An option that comes into existence when the
price of the underlying asset declines to a prespecified level.
Down-and-out Option An option that ceases to exist when the price of
the underlying asset declines to a prespecified level.
Downgrade Trigger A clause in a contract that states that the contract
can be terminated by one side if the credit rating of the other side falls
below a certain level.
Duration A measure of the average life a bond. It is also an approximation to the ratio of the proportional change in the bond price to the
absolute change in its yield.
Duration Matching A procedure for matching the durations of assets
and liabilities.
Dynamic Hedging A procedure for hedging an option position by
periodically changing the position held in the underlying asset. The
objective is usually to maintain a delta-neutral position.
EAD See Exposure at Default.
Early Exercise Exercise prior to the maturity date.
Economic Capital The capital that a bank's own calculation indicates it
needs.
Efficient Frontier The optimal trade-offs for an investor between
expected return and standard deviation of return.
Efficient Market Hypothesis A hypothesis that asset prices reflect
relevant information.
Electronic Trading System of trading where a computer is used to match
buyers and sellers.
Embedded Option An option that is an inseparable part of another
instrument.
Empirical Research Research based on historical market data.
Glossary of Terms
465
Equity Swap A swap where the return on an equity portfolio is
exchanged for either a fixed or a floating rate of interest.
Eurocurrency A currency that is outside the formal control of the
issuing country's monetary authorities.
Eurodollar
A dollar held in a bank outside the United States.
Eurodollar Futures Contract
deposit.
Eurodollar Interest Rate
European Option
life.
EWMA
A futures contract written on a Eurodollar
The interest rate on a Eurodollar deposit.
An option that can be exercised only at the end of its
Exponentially weighted moving average.
Exchange-traded Market Market organized by an exchange such as the
New York Stock Exchange or Chicago Board Options Exchange.
Ex-dividend Date When a dividend is declared, an ex-dividend date is
specified. Investors who own shares of the stock just before the exdividend date receive the dividend.
Exercise Price The price at which the underlying asset may be bought or
sold in an option contract. (Also called the strike price.)
Exotic Option
A nonstandard option.
Expectations Theory The theory that forward interest rates equal
expected future spot interest rates.
Expected Shortfall Expected loss during N days conditional on being in
the (100 — X)% tail of the distribution of profits/losses. The variable N is
the time horizon and X% is the confidence level.
Expected Value of a Variable The average value of the variable obtained
by weighting the alternative values by their probabilities.
Expiration Date
The end of life of a contract.
Exponentially Weighted Moving Average Model A model where exponential weighting is used to provide forecasts for a variable from historical
data. It is sometimes applied to variances and covariances in value-at-risk
calculations.
Exponential Weighting A weighting scheme where the weight given to
an observation depends on how recent it is. The weight given to an
observation t time periods ago is times the weight given to an observation t — 1 time periods ago, where < 1.
Glossary of Terms
466
Exposure at Default The maximum amount that could be lost (assuming no recovery) when a default occurs.
Extreme Value Theory A theory enabling the shape of the tails of a
distribution to be estimated from data.
Factor
Source of uncertainty.
Factor Analysis An analysis aimed at finding a small number of factors
that describe most of the variation in a large number of correlated
variables. (Similar to a principal components analysis.)
Factor Loadings The values of variables in a factor model when we have
one unit of a particular factor and no units of other factors.
Factor Model Model where a set of correlated variables are assumed to
depend linearly on a number of uncorrelated factors.
Factor Scores In a factor model this is the amount of different factors
present in a particular observation on the variables.
Financial Intermediary A bank or other financial institution that facilitates the flow of funds between different entities in the economy.
Floor
See Interest Rate Floor.
Floor-Ceiling Agreement
Floorlet
See Collar.
One component of a floor.
Floor Rate
The rate in an interest rate floor agreement.
Foreign Currency Option
An option on a foreign exchange rate.
Forward Contract A contract that obligates the holder to buy or sell an
asset for a predetermined delivery price at a predetermined future time.
Forward Exchange Rate
currency.
The forward price of one unit of a foreign
Forward Interest Rate The interest rate for a future period of time
implied by the rates prevailing in the market today.
Forward Price The delivery price in a forward contract that causes the
contract to be worth zero.
Forward Rate
rate.
Can refer to a forward interest rate or a forward exchange
Forward Rate Agreement (FRA) Agreement that a certain interest rate
will apply to a certain principal amount for a certain time period in the
future.
Glossary of Terms
467
Futures Contract A contract that obligates the holder to buy or sell an
asset at a predetermined delivery price during a specified future time
period. The contract is settled daily.
Futures Option
Futures Price
contract.
An option on a futures contract.
The delivery price currently applicable to a futures
G-30 Policy Recommendations A set of recommendations concerning
derivatives issued by nonregulators in 1993.
Gamma
The rate of change of delta with respect to the asset price.
Gamma-neutral portfolio
A portfolio with a gamma of zero.
GARCH Model A model for forecasting volatility where the variance
rate follows a mean-reverting process.
Gaussian Copula Model
normal distribution.
A copula model based on the multivariate
Glass-Steagall Act An act passed in the United States separating commercial and investment banks.
Greeks
Hedge parameters such as delta, gamma, vega, theta, and rho.
Haircut Discount applied to the value of an asset when it is used as
collateral.
Hazard Rate Measures probability of default in a short period of time
conditional on no earlier default.
HDD Heating degree days. The maximum of zero and the amount by
which the daily average temperature is less than 65° Fahrenheit. The
average temperature is the average of the highest and lowest temperatures
(midnight to midnight).
Hedge
A trade designed to reduce risk.
Hedge Funds Funds that are subject to less restrictions and less regulation than mutual funds. They can take short positions and use derivatives,
but they cannot publicly offer their securities.
Hedger
An individual who enters into hedging trades.
Hedge Ratio The ratio of the size of a position in a hedging instrument
to the size of the position being hedged.
Historical Simulation
Historic Volatility
A simulation based on historical data.
A volatility estimated from historical data.
Glossary of Terms
468
Holiday Calendar Calendar defining which days are holidays for the
purposes of determining payment dates in a financial transaction.
Hybrid Approach
capital.
Approach to aggregating different types of economic
Implied Volatility Volatility implied from an option price using the
Black-Scholes or a similar model.
Inception Profit Profit created by selling a derivative for more than its
theoretical value.
Incremental Value at Risk The difference between the value at risk with
and without a particular component of the portfolio.
Initial Margin
trade.
The cash required from a futures trader at the time of the
Instantaneous Forward Rate
time in the future.
Forward rate for a very short period of
Interest Rate Cap An option that provides a payoff when a specified
interest rate is above a certain level. The interest rate is a floating rate that
is reset periodically.
Interest Rate Collar
interest rate floor.
A combination of an interest rate cap and an
Interest Rate Derivative
future interest rates.
A derivative whose payoffs are dependent on
Interest Rate Floor An option that provides a payoff when an interest
rate is below a certain level. The interest rate is a floating rate that is reset
periodically.
Interest Rate Option
level of interest rates.
An option where the payoff is dependent on the
Interest Rate Swap An exchange of a fixed rate of interest on a certain
notional principal for a floating rate of interest on the same notional
principal.
In-the-money Option Either (a) a call option where the asset price is
greater than the strike price or (b) a put option where the asset price is
less than the strike price.
Intrinsic Value For a call option, this is the greater of the excess of the
asset price over the strike price and zero. For a put option, it is the greater
of the excess of the strike price over the asset price and zero.
Glossary of Terms
469
Investment Asset An asset held by significant numbers of individuals
for investment purposes.
iTraxx An index of the credit quality of 125 European investment grade
companies.
IRB Approach Internal ratings based approach for assessing credit risk
capital in Basel II.
Key Risk Indicators Indicators to track the level of operational risk.
Kurtosis A measure of the fatness of the tails of a distribution.
LGD See Loss Given Default.
LIBID London interbank bid rate. The rate bid by banks on Eurocurrency deposits (i.e., the rate at which a bank is willing to borrow from
other banks).
LIBOR London interbank offered rate. The rate offered by banks on
Eurocurrency deposits (i.e., the rate at which a bank is willing to lend to
other banks).
LIBOR-in-arrears Swap Swap where the interest paid on a date is
determined by the interest rate observed on that date (not by the interest
rate observed on the previous payment date).
LIBOR/Swap Zero Curve Zero rates as a function of maturity that are
calculated from LIBOR rates, eurodollar futures, and swap rates.
LIBOR Zero Curve See LIBOR/Swap Zero Curve.
Linear Product Derivative product whose price depends linearly on one
or more underlying variables.
Liquidity-adjusted VaR A value-at-risk calculation that takes account of
the impact of the bid-offer spread when positions are unwound.
Liquidity Black Holes The risk that liquidity will dry up because everyone wants to be on the same side of the market.
Liquidity Preference Theory A theory leading to the conclusion that
forward interest rates are above expected future spot interest rates.
Liquidity Premium The amount that forward interest rates exceed
expected future spot interest rates.
Liquidity Risk Risk that it will not be possible to sell a holding of a
particular instrument at its theoretical price.
Lognormal Distribution A variable has a lognormal distribution when
the logarithm of the variable has a normal distribution.
Long Position A position involving the purchase of an asset.
470
Glossary of Terms
Lookback Option An option whose payoff is dependent on the maximum or minimum of the asset price achieved during a certain period.
Loss Given Default The percentage of the exposure to a counterparty
that is lost when a default by the counterparty occurs.
Maintenance Margin When the balance in a trader's margin account
falls below the maintenance margin level, the trader receives a margin call
requiring the account to be topped up to the initial margin level.
Margin The cash balance (or security deposit) required from a futures
or options trader.
Margin Call A request for extra margin when the balance in the margin
account falls below the maintenance margin level.
Marginal Value at Risk The rate of change of the value at risk with the
size of one component of the portfolio.
Market Maker A trader who is willing to quote both bid and offer prices
for an asset.
Market Model A model most commonly used by traders.
Market Portfolio A portfolio consisting of the universe of all possible
investments.
Market Risk Risk relating to movements in market variables.
Marking to Market The practice of revaluing an instrument to reflect the
current values of the relevant market variables.
Maturity Date The end of the life of a contract.
Maximum-likelihood Method A method for choosing the values of
parameters by maximizing the probability of a set of observations
occurring.
Mean Reversion The tendency of a market variable (such as a volatility
or an interest rate) to revert back to some long-run average level.
Merton's Model Model using equity prices to estimate default probabilities. (Other models developed by Merton are also sometimes referred
to as Merton's model.)
Model-building Approach The use of a model to estimate value at risk.
Model Risk The risk relating to the use of models to price derivative
products.
Modified Duration A modification to the standard duration measure so
that it more accurately describes the relationship between proportional
changes in a bond price and actual changes in its yield. The modification
Glossary of Terms
471
takes account of the compounding frequency with which the yield is
quoted.
Monte Carlo Simulation A procedure for randomly sampling changes in
market variables.
Moral Hazard The possibility that the behavior of an insured entity will
change because of the existence of an insurance contract.
Multivariate Normal Distribution
ables, each of which is normal.
The joint distribution of many vari-
Naked Position A short position in a call option that is not combined
with a long position in the underlying asset.
Net Interest Income
a bank.
The excess of interest earned over interest paid for
Net Replacement Ratio The ratio of current exposure with netting to
current exposure without netting.
Netting The ability to offset contracts with positive and negative values
in the event of a default by a counterparty.
Nonlinear product Derivative product that is not linearly dependent on
the underlying variables.
Nonsystematic risk
Risk that can be diversified away.
Normal Distribution
Normal Market
The standard bell-shaped distribution of statistics.
A market where futures prices increase with maturity.
Notional Principal The principal used to calculate payments in an
interest rate swap. The principal is "notional" because it is neither paid
nor received.
Numerical Procedure
available.
A method of calculation when no formula is
Offer Price The price that a dealer is offering to sell an asset. (Also
called the ask price.)
Open Interest The total number of long positions outstanding in a
futures contract (equals the total number of short positions).
Open Outcry
exchange.
System of trading where traders meet on the floor of the
Operational Risk
The risk of loss arising from inadequate or failed
internal processes, people, and systems, or from external events.
Option
The right to buy or sell an asset.
Glossary of Terms
472
Out-of-the-money Option Either (a) a call option where the asset price
is less than the strike price or (b) a put option where the asset price is
greater than the strike price.
Over-the-counter Market A market where traders deal by phone. The
traders are usually financial institutions, corporations, and fund managers.
Par Value
The principal amount of a bond.
Par Yield
principal.
The coupon on a bond that makes its price equal the
Parallel Shift A movement in the yield curve where each point on the
curve changes by the same amount.
Partial Duration Percentage change in value of a portfolio for a small
change in one point on the zero-coupon yield curve.
Payoff The cash realized by the holder of an option or other derivative
at the end of its life.
PD
Probability of default.
Plain Vanilla
A term used to describe a standard deal.
Poisson Distribution Distribution for number of events in a certain time
period in a Poisson process.
Poisson Process A process describing a situation where events happen at
random. The probability of an event in time
where is the
intensity of the process.
Portfolio Immunization
interest rates.
Making a portfolio relatively insensitive to
Portfolio Insurance Entering into trades to ensure that the value of a
portfolio will not fall below a certain level.
Positive SemiDefinite Condition that must be satisfied by a Variancecovariance matrix for it to be valid.
Power Law Law describing the tails of many probability distributions
that are encountered in practice.
Premium
The price of an option.
Principal
The par or face value of a debt instrument.
Principal Components Analysis An analysis aimed at finding a small
number of factors that describe most of the variation in a large number of
correlated variables. (Similar to a factor analysis.)
Glossary of Terms
473
Put-Call Parity The relationship between the price of a European call
option and the price of a European put option when they have the same
strike price and maturity date.
Put Option An option to sell an asset for a certain price by a certain
date.
Puttable Bond A bond where the holder has the right to sell it back to
the issuer at certain predetermined times for a predetermined price.
Puttable Swap
A swap where one side has the right to terminate early.
Quadratic Model Quadratic relationship between change in portfolio
value and percentage changes in market variables.
Quantitative Impact Studies Studies by the Basel Committee of the
effect of proposed new regulations on capital of banks.
RAROC
Risk-adjusted return on capital.
Rebalancing The process of adjusting a trading position periodically.
Usually the purpose is to maintain delta neutrality.
Recovery Rate Amount recovered in the event of a default as a percentage of the face value.
Regulatory arbitrage Transactions designed to reduce the total regulatory capital of the financial institutions involved.
Regulatory capital
lators to keep.
Capital a financial institution is required by regu-
Repo Repurchase agreement. A procedure for borrowing money by
selling securities to a counterparty and agreeing to buy them back later
at a slightly higher price.
Repo Rate
The rate of interest in a repo transaction.
Reset Date The date in a swap or cap or floor when the floating rate for
the next period is set.
Reversion Level The level that the value of a market variable (e.g., a
volatility) tends to revert.
Rho
Rate of change of the price of a derivative with the interest rate.
Risk-free Rate
any risks.
The rate of interest that can be earned without assuming
Risk-neutral Valuation The valuation of an option or other derivative
assuming the world is risk neutral. Risk-neutral valuation gives the
correct price for a derivative in all worlds, not just in a risk-neutral world.
474
Glossary of Terms
Risk-neutral World A world where investors are assumed to require no
extra return on average for bearing risks.
Risk-weighted Amount
See Risk-weighted Assets.
Risk-weighted Assets Quantity calculated in Basel I and Basel II. Total
capital must be at least 8% of risk-weighted assets.
Roll Back
See Backwards Induction.
Sarbanes-Oxley Act passed in the United States in 2002 increasing the
responsibilities of directors, CEOs, and CFOs of public companies.
Scenario Analysis An analysis of the effects of possible alternative
future movements in market variables on the value of a portfolio. Also
used to generate scenarios leading to operational risk losses.
Scorecard Approach
risk.
SEC
A self-assessment procedure used for operational
Securities and Exchange Commission.
Short Position
own.
A position assumed when traders sell shares they do not
Short Selling Selling in the market shares that have been borrowed from
another investor.
Simulation
See Monte Carlo Simulation.
Solvency II A new regulatory framework for insurance companies
proposed by the European Union.
Specific Risk Charge
trading book.
Capital requirement for idiosyncratic risks in the
Spectral Risk Measure Risk measure that assigns weights to the quantiles
of the loss distribution.
Speculator An individual who is taking a position in the market.
Usually the individual is betting that the price of an asset will go up or
that the price of an asset will go down.
Spot Interest Rate
Spot Price
See Zero-coupon Interest Rate.
The price for immediate delivery.
Spot Volatilities The volatilities used to price a cap when a different
volatility is used for each caplet.
Static Hedge
initiated.
A hedge that does not have to be changed once it is
475
Glossary of Terms
Static Options Replication A procedure for hedging a portfolio that
involves finding another portfolio of approximately equal value on some
boundary.
Stochastic Variable Variable whose future value is uncertain.
Stock Index
An index monitoring the value of a portfolio of stocks.
Stock Index Futures
Futures on a stock index.
Stock Index Option
An option on a stock index.
Stock Option
Option on a stock.
Storage Costs
The costs of storing a commodity.
Stress Testing Testing of the impact of extreme market moves on the
value of a portfolio.
Strike Price
The price at which the asset may be bought or sold in an
option contract. (Also called the exercise price.)
Structured Product
Derivative designed by a financial institution to
meet the needs of a client.
Student t-Copula Copula based on the multivariate Student t-distribuStudent t-Distribution
Distribution with heavier tails than the normal
distribution.
Swap An agreement to exchange cash flows in the future according to a
prearranged formula.
Swap Rate The fixed rate in an interest rate swap that causes the swap to
have a value of zero.
Swap Zero Curve
See LIBOR/Swap Zero Curve.
Swaption An option to enter into an interest rate swap where a specified
fixed rate is exchanged for floating.
Synthetic CDO
Synthetic Option
Systematic Risk
A CDO created by selling credit default swaps.
An option created by trading the underlying asset.
Risk that cannot be diversified away.
Systemic Risk Risk that a default by one financial institution will lead to
defaults by other financial institutions.
Tail Correlation Correlation between the tails of two distributions.
Measures the extent to which extreme values tend to occur together.
Tail Loss
See Expected Shortfall.
Glossary of Terms
476
Taylor Series Expansion For a function of several variables, this relates
changes in the value of the function to changes in the values of the
variables when the changes are small.
Term Structure of Interest Rates
and their maturities.
Terminal Value
The relationship between interest rates
The value at maturity.
Theta The rate of change of the price of an option or other derivative
with the passage of time.
Tier 1 Capital
Equity and similar sources of capital.
Tier 2 Capital Subordinated debt (life greater than five years) and
similar sources of capital.
Tier 3 Capital
years).
Time Decay
Short-term subordinated debt (life between two and five
See Theta.
Time Value The value of an option arising from the time left to
maturity (equals an option's price minus its intrinsic value).
Total Return Swap A swap where the return on an asset such as a bond
is exchanged for LIBOR plus a spread. The return on the asset includes
income such as coupons and the change in value of the asset.
Trading Days
Days when markets are open for trading.
Tranche One of several securities that have different risk attributes.
Examples are the tranches of a CDO.
Transaction Costs The cost of carrying out a trade (commissions plus
the difference between the price obtained and the midpoint of the bidoffer spread).
Treasury Bill A short-term non-coupon-bearing instrument issued by
the government to finance its debt.
Treasury Bond A long-term coupon-bearing instrument issued by the
government to finance it debt.
Treasury Note
Treasury bond lasting less than 10 years.
Treasury Note Futures
A futures contract on Treasury notes.
Tree Representation of the evolution of the value of a market variable
for the purposes of valuing an option or other derivative.
Underlying Variable A variable that the price of an option or other
derivative depends on.
Glossary of Terms
All
Unsystematic risk See Nonsystematic Risk.
Up-and-in Option An option that comes into existence when the price
of the underlying asset increases to a prespecified level.
Up-and-out Option An option that ceases to exist when the price of the
underlying asset increases to a prespecified level.
Value at Risk A loss that will not be exceeded at some specified
confidence level.
Variance-covariance matrix A matrix showing variances of, and covariances between, a number of different market variables.
Variance Rate
The square of volatility.
Vasicek's Model Model of default correlation based on the Gaussian
copula. (Other models developed by Vasicek are also sometimes referred
to as Vasicek's model.)
Vega The rate of change in the price of an option or other derivative
with volatility.
Vega-neutral Portfolio A portfolio with a vega of zero.
Volatility A measure of the uncertainty of the return realized on an
asset.
Volatility Skew A term used to describe the volatility smile when it is
nonsymmetrical.
Volatility Smile
The variation of implied volatility with strike price.
Volatility Surface A table showing the variation of implied volatilities
with strike price and time to maturity.
Volatility Term Structure
to maturity.
The variation of implied volatility with time
Weather Derivative Derivative where the payoff depends on the
weather.
Writing an Option Selling an option.
Yield A return provided by an instrument.
Yield Curve See Term Structure.
Zero-coupon Bond A bond that provides no coupons.
Zero-coupon Interest Rate The interest rate that would be earned on a
bond that provides no coupons.
Zero-coupon Yield Curve A plot of the zero-coupon interest rate
against time to maturity.
478
Glossary of Terms
Zero Curve See Zero-coupon Yield Curve.
Zero Rate See Zero-coupon Interest Rate.
Z-Score A number indicating how likely a company is to default.
DerivaGem Software
You can download the DerivaGem option calculator from the author's
website:
http://www.rotman.utoronto.ca/~hull
The software requires Microsoft Windows 98 or later and Microsoft Excel
2000 or later. It consists of two files: DG151.dll and DG151.xls. To install
the software, you should create a folder with the name DerivaGem (or
some other name of your own choosing) and load the files into the folder.
You MUST then move DG151.dll into the Windows\System folder or the
Windows\System 32 folder.1
Users should ensure that Security for Macros in Excel is set at Medium
or Low. Check Tools followed by Macros followed by Security in Excel to
change this. While using the software, you may be asked whether you
want to enable macros. You should click Enable Macros.
THE OPTIONS CALCULATOR
DG151.xls is a user-friendly option calculator. It consists of three work-sheets. The first worksheet is used to carry out computations for stock
1
Note that it is not uncommon for Windows Explorer to be set up so that *.dll files are
not displayed. To change the setting so that the *.dll file can be seen proceed as follows.
In Windows 98 and ME, click View, followed by File Options, followed by View, followed
by Show All Files. In Windows 2000, XP, and NT, click Tools, followed by Folder
Options, followed by View, followed by Show Hidden Files and Folders.
480
DerivaGem
Software
options, currency options, index options, and futures options; the second
is used for European and American bond options; and the third is used
for caps, floors, and European swap options.
The software produces prices, Greek letters, and implied volatilities for
a wide range of different instruments. It displays charts showing the way
that option prices and the Greek letters depend on inputs. It also
displays binomial and trinomial trees showing how the computations
are carried out.
General Operation
To use the option calculator, you should choose a worksheet and click on
the appropriate buttons to select Option Type, Underlying Type, and so
on. You should then enter the parameters for the option you are
considering, hit Enter on your keyboard, and click on Calculate. DerivaGem will then display the price or implied volatility for the option you
are considering together with Greek letters. If the price has been calculated from a tree, and you are using the first or second worksheet, you can
then click on Display Tree to see the tree. If a tree is to be displayed, there
must be no more than ten time steps. An example of the tree that is
displayed is shown in Appendix D. Many different charts can be displayed in all three worksheets. To display a chart, you must first choose
the variable you require on the vertical axis, the variable you require on
the horizontal axis, and the range of values to be considered on the
horizontal axis. Following that you should hit Enter on your keyboard
and click on Draw Graph. Whenever the values in one or more cells are
changed, it is necessary to hit Enter on your keyboard before clicking on
one of the buttons.
You may be asked whether you want to update to the new version
when you first save the software. You should choose the Yes button.
Options on Stocks, Currencies, Indices, and Futures
The first worksheet (Equity_FX_Index_Futures) is used for options on
stocks, currencies, indices, and futures. To use it, you should first select
the Underlying Type (Equity, Currency, Index, or Futures). You should
then select the Option Type. The alternatives are: Analytic European (i.e.,
Black-Scholes for a European option), Binomial European (i.e., European option using a binomial tree), Binomial American (i.e., American
option using a binomial tree), Asian, Barrier Up and In, Barrier Up and
Out, Barrier Down and In, Barrier Down and Out, Binary Cash or
DerivaGem
Software
481
Nothing, Binary Asset or Nothing, Chooser, Compound Option on Call,
Compound Option on Put, or Lookback. You should then enter the data
on the underlying asset and the data on the option. Note that all interest
rates are expressed with continuous compounding.
In the case of European and American equity options, a table pops up
allowing you to enter dividends. Enter the time of each ex-dividend date
(measured in years from today) in the first column and the amount of the
dividend in the second column. Dividends must be entered in chronological order.
You must click on buttons to choose whether the option is a call or a
put and whether you wish to calculate an implied volatility. If you do
wish to calculate an implied volatility, the option price should be entered
in the cell labeled Price.
Once all the data has been entered you should hit Enter on your
keyboard and click on Calculate. If Implied Volatility was selected,
DerivaGem displays the implied volatility in the Volatility (% per year)
cell. If Implied Volatility was not selected, it uses the volatility you
entered in this cell and displays the option price in the Price cell.
Once the calculations have been completed, the tree (if used) can be
inspected and charts can be displayed.
When Analytic European is selected, DerivaGem uses the equations in
Appendix C to calculate prices and Greek letters. When Binomial European or Binomial American is selected, a binomial tree is constructed as
described in Appendix D. Up to 500 time steps can be used.
The input data are largely self-explanatory. In the case of an Asian
option, the Current Average is the average price since inception. If the
Asian option is new (Time since Inception equals zero), then the Current
Average cell is irrelevant and can be left blank. In the case of a Lookback
Option, the Minimum to Date is used when a Call is valued and the
Maximum to Date is used when a Put is valued. For a new deal, these
should be set equal to the current price of the underlying asset.
Bond Options
The second worksheet (Bond_Options) is used for European and
American options on bonds. You should first select a pricing model
(Black-European, Normal-Analytic European, Normal-Tree European,
Normal-American, Lognormal-European, or Lognormal-American;
these models are explained in John Hull's book Options, Futures, and
Other Derivatives). You should then enter the Bond Data and the Option
482
DerivaGem
Software
Data. The coupon is the rate paid per year and the frequency of payments
can be selected as Quarterly, Semi-Annual, or Annual. The zero-coupon
yield curve is entered in the table labeled Term Structure. Enter maturities
(measured in years) in the first column and the corresponding continuously compounded rates in the second column. The maturities should
be entered in chronological order. DerivaGem assumes a piecewise linear
zero curve similar to that in Figure 4.1. Note that, when valuing interest
rate derivatives, DerivaGem rounds all times to the nearest whole number
of days.
When all data have been entered, hit Enter on your keyboard. The
quoted bond price per $100 of principal, calculated from the zero curve, is
displayed when the calculations are complete. You should indicate
whether the option is a call or a put and whether the strike price is a
quoted (clean) strike price or a cash (dirty) strike price. (The cash price is
the quoted price plus accrued interest.) Note that the strike price is
entered as the price per $100 of principal. You should indicate whether
you are considering a call or a put option and whether you wish to
calculate an implied volatility. If you select implied volatility and the
normal model or lognormal model is used, DerivaGem implies the shortrate volatility keeping the reversion rate fixed.
Once all the inputs are complete, you should hit Enter on your keyboard and click Calculate. After that, the tree (if used) can be inspected
and charts can be displayed. Note that the tree displayed lasts until the
end of the life of the option. DerivaGem uses a much larger tree in its
computations to value the underlying bond.
Note that Black's model is similar to Black-Scholes and assumes that
the bond price is lognormal at option maturity. The approximate duration relationship in Chapter 4 is used to convert bond yield volatilities to
bond price volatilities. This is the usual market practice.
Caps and Swap Options
The third worksheet (Caps_and_Swap_Options) is used for caps and
swap options. You should first select the Option Type (Swap Option or
Cap/Floor) and Pricing Model (Black-European, Normal-European, or
Lognormal-European; these products and the alternative models are
explained in John Hull's book Options, Futures, and Other Derivatives).
You should then enter data on the option you are considering. The
Settlement Frequency indicates the frequency of payments and can be
Annual, Semi-Annual, Quarterly, or Monthly. The software calculates
DerivaGem
Software
483
payment dates by working backward from the end of the life of the cap or
swap option. The initial accrual period may be a nonstandard length
between 0.5 and 1.5 times a normal accrual period. The software can be
used to imply either a volatility or a cap rate/swap rate from the price.
When a normal model or a lognormal model is used, DerivaGem implies
the short-rate volatility keeping the reversion rate fixed. The zero-coupon
yield curve is entered in the table labeled Term Structure. Enter maturities
(measured in years) in the first column and the corresponding continuously compounded rates in the second column. The maturities should be
entered in chronological order. DerivaGem assumes a piecewise linear
zero curve similar to that in Figure 4.1.
Once all the inputs are complete, you should click Calculate. After that,
charts can be displayed. Note that when Black's model is used, DerivaGem assumes (a) that future interest rates are lognormal when caps are
valued and (b) that future swap rates are lognormal when swap options
are valued.
Greek Letters
In the Equity_FX_Index_Futures worksheet, the Greek letters are calculated as follows.
Delta: Change in option price per dollar increase in underlying
asset.
Gamma: Change in delta per dollar increase in underlying asset.
Vega: Change in option price per 1% increase in volatility (e.g.,
volatility increases from 20% to 21%).
Rho: Change in option price per 1% increase in interest rate (e.g.,
interest increases from 5% to 6%>).
Theta: Change in option price per calendar day passing.
In the Bond_Options and Caps_and_Swap_Options worksheets, the
Greek letters are calculated as follows:
DV01: Change in option price per one basis point upward
parallel shift in the zero curve.
Gamma01: Change in DV01 per one basis point upward parallel shift
in the zero curve, multiplied by 100.
Vega: Change in option price when volatility parameter increases
by 1%) (e.g., volatility increases from 20% to 21 %)
Table for N(x) When x
0
This table shows values of N(x) for x 0. The table should be used with interpolation.
For example,
N(-0.1234) = N(-0.12) - 0.34[N(-0.12) - N(-0.13)]
= 0.4522 - 0.34 x (0.4522 - 0.4483)
= 0.4509
X
-0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1.0
-1.1
-1.2
-1.3
-1.4
-1.5
-1.6
-1.7
-1.8
-1.9
-2.0
-2.1
-2.2
-2.3
-2.4
-2.5
-2.6
-2.7
-2.8
-2.9
-3.0
-3.1
-3.2
-3.3
-3.4
-3.5
-3.6
-3.7
-3.8
-3.9
-4.0
.00
.01
.02
.03
.04
.05
.06
.07
.08
. 09
0.5000
0.4602
0.4207
0.3821
0.3446
0.3085
0.2743
0.2420
0.2119
0.1841
0.1587
0.1357
0.1151
0.0968
0.0808
0.0668
0.0548
0.0446
0.0359
0.0287
0.0228
0.0179
0.0139
0.0107
0.0082
0.0062
0.0047
0.0035
0.0026
0.0019
0.0014
0.0010
0.0007
0.0005
0.0003
0.0002
0.0002
0.0001
0.0001
0.0000
0.0000
0.4960
0.4562
0.4168
0.3783
0.3409
0.3050
0.2709
0.2389
0.2090
0.1814
0.1562
0.1335
0.1131
0.0951
0.0793
0.0655
0.0537
0.0436
0.0351
0.0281
0.0222
0.0174
0.0136
0.0104
0.0080
0.0060
0.0045
0.0034
0.0025
0.0018
0.0013
0.0009
0.0007
0.0005
0.0003
0.0002
0.0002
0.0001
0.0001
0.0000
0.0000
0.4920
0.4522
0.4129
0.3745
0.3372
0.3015
0.2676
0.2358
0.2061
0.1788
0.1539
0.1314
0.1112
0.0934
0.0778
0.0643
0.0526
0.0427
0.0344
0.0274
0.0217
0.0170
0.0132
0.0102
0.0078
0.0059
0.0044
0.0033
0.0024
0.0018
0.0013
0.0009
0.0006
0.0005
0.0003
0.0002
0.0001
0.0001
0.0001
0.0000
0.0000
0.4880
0.4483
0.4090
0.3707
0.3336
0.2981
0.2643
0.2327
0.2033
0.1762
0.1515
0.1292
0.1093
0.0918
0.0764
0.0630
0.0516
0.0418
0.0336
0.0268
0.0212
0.0166
0.0129
0.0099
0.0075
0.0057
0.0043
0.0032
0.0023
0.0017
0.0012
0.0009
0.0006
0.0004
0.0003
0.0002
0.0001
0.0001
0.0001
0.0000
0.0000
0.4840
0.4443
0.4052
0.3669
0.3300
0.2946
0.2611
0.2296
0.2005
0.1736
0.1492
0.1271
0.1075
0.0901
0.0749
0.0618
0.0505
0.0409
0.0329
0.0262
0.0207
0.0162
0.0125
0.0096
0.0073
0.0055
0.0041
0.0031
0.0023
0.0016
0.0012
0.0008
0.0006
0.0004
0.0003
0.0002
0.0001
0.0001
0.0001
0.0000
0.0000
0.4801
0.4404
0.4013
0.3632
0.3264
0.2912
0.2578
0.2266
0.1977
0.1711
0.1469
0.1251
0.1056
0.0885
0.0735
0.0606
0.0495
0.0401
0.0322
0.0256
0.0202
0.0158
0.0122
0.0094
0.0071
0.0054
0.0040
0.0030
0.0022
0.0016
0.0011
0.0008
0.0006
0.0004
0.0003
0.0002
0.0001
0.0001
0.0001
0.0000
0.0000
0.4761
0.4364
0.3974
0.3594
0.3228
0.2877
0.2546
0.2236
0.1949
0.1685
0.1446
0.1230
0.1038
0.0869
0.0721
0.0594
0.0485
0.0392
0.0314
0.0250
0.0197
0.0154
0.0119
0.0091
0.0069
0.0052
0.0039
0.0029
0.0021
0.0015
0.0011
0.0008
0.0006
0.0004
0.0003
0.0002
0.0001
0.0001
0.0001
0.0000
0.0000
0.4721
0.4325
0.3936
0.3557
0.3192
0.2843
0.2514
0.2206
0.1922
0.1660
0.1423
0.1210
0.1020
0.0853
0.0708
0.0582
0.0475
0.0384
0.0307
0.0244
0.0192
0.0150
0.0116
0.0089
0.0068
0.0051
0.0038
0.0028
0.0021
0.0015
0.0011
0.0008
0.0005
0.0004
0.0003
0.0002
0.0001
0.0001
0.0001
0.0000
0.0000
0.4681
0.4286
0.3897
0.3520
0.3156
0.2810
0.2483
0.2177
0.1894
0.1635
0.1401
0.1190
0.1003
0.0838
0.0694
0.0571
0.0465
0.0375
0.0301
0.0239
0.0188
0.0146
0.0113
0.0087
0.0066
0.0049
0.0037
0.0027
0.0020
0.0014
0.0010
0.0007
0.0005
0.0004
0.0003
0.0002
0.0001
0.0001
0.0001
0.0000
0.0000
0.4641
0.4247
0.3859
0.3483
0.3121
0.2776
0.2451
0.2148
0.1867
0.1611
0.1379
0.1170
0.0985
0.0823
0.0681
0.0559
0.0455
0.0367
0.0294
0.0233
0.0183
0.0143
0.0110
0.0084
0.0064
0.0048
0.0036
0.0026
0.0019
0.0014
0.0010
0.0007
0.0005
0.0003
0.0002
0.0002
0.0001
0.0001
0.0001
0.0000
0.0000
Table for N(x) W h e n x
This table shows values of N{x) for x
For example,
N(0.6278) = N(0.62) + 0.78[N(0.63) - N(0.62)]
= 0.7324 + 0.78 x (0.7357 - 0.7324)
= 0.7350
X
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
0.5000
0.5398
0.5793
0.6179
0.6554
0.6915
0.7257
0.7580
0.7881
0.8159
0.8413
0.8643
0.8849
0.9032
0.9192
0.9332
0.9452
0.9554
0.9641
0.9713
0.9772
0.9821
0.9861
0.9893
0.9918
0.9938
0.9953
0.9965
0.9974
0.9981
0.9986
0.9990
0.9993
0.9995
0.9997
0.9998
0.9998
0.9999
0.9999
1.0000
1.0000
0.5040
0.5438
0.5832
0.6217
0.6591
0.6950
0.7291
0.7611
0.7910
0.8186
0.8438
0.8665
0.8869
0.9049
0.9207
0.9345
0.9463
0.9564
0.9649
0.9719
0.9778
0.9826
0.9864
0.9896
0.9920
0.9940
0.9955
0.9966
0.9975
0.9982
0.9987
0.9991
0.9993
0.9995
0.9997
0.9998
0.9998
0.9999
0.9999
1.0000
1.0000
0.5080
0.5478
0.5871
0.6255
0.6628
0.6985
0.7324
0.7642
0.7939
0.8212
0.8461
0.8686
0.8888
0.9066
0.9222
0.9357
0.9474
0.9573
0.9656
0.9726
0.9783
0.9830
0.9868
0.9898
0.9922
0.9941
0.9956
0.9967
0.9976
0.9982
0.9987
0.9991
0.9994
0.9995
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
1.0000
0.5120
0.5517
0.5910
0.6293
0.6664
0.7019
0.7357
0.7673
0.7967
0.8238
0.8485
0.8708
0.8907
0.9082
0.9236
0.9370
0.9484
0.9582
0.9664
0.9732
0.9788
0.9834
0.9871
0.9901
0.9925
0.9943
0.9957
0.9968
0.9977
0.9983
0.9988
0.9991
0.9994
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
1.0000
0.5160
0.5557
0.5948
0.6331
0.6700
0.7054
0.7389
0.7704
0.7995
0.8264
0.8508
0.8729
0.8925
0.9099
0.9251
0.9382
0.9495
0.9591
0.9671
0.9738
0.9793
0.9838
0.9875
0.9904
0.9927
0.9945
0.9959
0.9969
0.9977
0.9984
0.9988
0.9992
0.9994
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
1.0000
0.5199
0.5596
0.5987
0.6368
0.6736
0.7088
0.7422
0.7734
0.8023
0.8289
0.8531
0.8749
0.8944
0.9115
0.9265
0.9394
0.9505
0.9599
0.9678
0.9744
0.9798
0.9842
0.9878
0.9906
0.9929
0.9946
0.9960
0.9970
0.9978
0.9984
0.9989
0.9992
0.9994
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
1.0000
0.5239
0.5636
0.6026
0.6406
0.6772
0.7123
0.7454
0.7764
0.8051
0.8315
0.8554
0.8770
0.8962
0.9131
0.9279
0.9406
0.9515
0.9608
0.9686
0.9750
0.9803
0.9846
0.9881
0.9909
0.9931
0.9948
0.9961
0.9971
0.9979
0.9985
0.9989
0.9992
0.9994
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
1.0000
0.5279
0.5675
0.6064
0.6443
0.6808
0.7157
0.7486
0.7794
0.8078
0.8340
0.8577
0.8790
0.8980
0.9147
0.9292
0.9418
0.9525
0.9616
0.9693
0.9756
0.9808
0.9850
0.9884
0.9911
0.9932
0.9949
0.9962
0.9972
0.9979
0.9985
0.9989
0.9992
0.9995
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
1.0000
0.5319
0.5714
0.6103
0.6480
0.6844
0.7190
0.7517
0.7823
0.8106
0.8365
0.8599
0.8810
0.8997
0.9162
0.9306
0.9429
0.9535
0.9625
0.9699
0.9761
0.9812
0.9854
0.9887
0.9913
0.9934
0.9951
0.9963
0.9973
0.9980
0.9986
0.9990
0.9993
0.9995
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
1.0000
0.5359
0.5753
0.6141
0.6517
0.6879
0.7224
0.7549
0.7852
0.8133
0.8389
0.8621
0.8830
0.9015
0.9177
0.9319
0.9441
0.9545
0.9633
0.9706
0.9767
0.9817
0.9857
0.9890
0.9916
0.9936
0.9952
0.9964
0.9974
0.9981
0.9986
0.9990
0.9993
0.9995
0.9997
0.9998
0.9998
0.9999
0.9999
0.9999
1.0000
1.0000
Index
Page numbers in bold refer to the Glossary of Terms
1996 Amendment, 176-78
Accrued interest, 457
Add-up basket CDS, 311
Add-on factor, 171, 457
Advanced IRB approach, 185-87,
257
Advanced measurement approach,
188, 324-26, 457
Adverse selection, 336, 457
Allen, S. L., 106
Allied Irish Bank, 321, 331, 395, 396
Altman, E.I., 257,261, 273
AMA, see Advanced measurement
approach
Amato, J. D., 269
American option, 40, 457
Analytic result, 457
Andersen, L., 317
APT, see Arbitrage pricing theory
Arbitrage, 457
Arbitrage pricing theory, 12, 457
Arbitrageur, 457
ARCH(m) model, 122
Arditti, R, 392
Artzner, P., 199, 214
Asian basket option, 48
Asian option, 46, 458
Ask price, 458
Asked price, 458
Asset swap, 264, 458
Asset swap spread, 264
Asset/liability management, 20-22
At-the-money option, 40, 458
Autocorrelation, 458
of volatility in GARCH models,
132
of daily profit/loss in VaR models
204
Average price call option, 46, 458
Average price put option, 458
Back office, 55, 400
Back testing, 208-11, 458
Backwards induction, 458
Bangia, A.,356
488
Bank capital, 15-18
regulatory, 165-94
economic, 365-81
Bank for International Settlements,
169
Bank of America, 322
Bankers Trust, 39, 401
Banking book, 176
Bankruptcy cost, 13-15, 29, 458
Barings Bank, 1, 49, 321, 322, 373,
396
Barrier option, 46, 458
Basak, S., 214
Basel Committee, 169, 210, 359
Basel I, 169, 458
Basel IA, 178
Basel II, 178-91, 359, 458
credit risk capital, 179-88
operational risk capital, 188—89,
324-36
Pillar 1, 179
Pillar 2, 179, 189
Pillar 3, 179, 190
market discipline, 179, 190
supervisory review, 179, 189
Basel III, 189
Basic indicator approach, 188, 324,
458
Basis, 458
Basis point, 91, 459
Basis risk, 43, 459
Basket credit default swap, 311,
314-16, 459
Basket option, 46, 459
Basu, S., 317
Baud, N., 339
Baz, J., 50
Beaglehole, D. R., 391, 393
Beder, T., 214
Beta, 10, 44, 459
Bid price, 28, 459
Bid-ask spread, 459
Bid-offer spread, 459
Index
Binary credit default swap, 307, 459
Binary option, 46, 459
Binomial correlation measure, 293-94
Binomial model, 459
Binomial tree, 418, 459
BIS Accord, 169-72
Bivariate normal distribution, 150,
156, 459
Bivariate Student t-distribution, 157
Black-Scholes-Merton model, 62,
347, 413, 459
Black's model, 459
Bollerslev, T., 125, 138
Bond option, 459
Bond,
clean price, 461
convexity, 93-94
coupon, 84
dirty price, 463
duration, 89-93
face value, 84
par value, 84
pricing, 84
principal, 84
yield, 84
Bond yield, 84, 459
Bootstrap method,
for calculating zero rates, 86-87,
459
for calculating VaR confidence
levels, 223
Boudoukh, J., 214, 221, 230
Box, G.E.P., 133
Boyle, P., 50
Boyle, F., 50
Brady, B., 261
Brown, K. C, 50
Brown, G.W., 50
Buckets, 99
Bunching, 211, 460
Business disruption and system
failures, 321, 327
Business risk, 368, 460
Index
•
1
Business risk economic capital, 370
Cai, L., 392
Calendar day, 113, 460
Calibration, 344, 460
Callable bond, 460
Call option, 40, 460
Cancelable swap, 460
Canter, M.S., 392
Cantor, R., 261
Cao, M., 392
Cap, 460
Cap rate, 460
Capital adequacy, 17-18
Capital asset pricing model, 3, 10,
460
Capital requirements directive, 167
Caplet, 460
CAPM, see Capital asset pricing
model
Cash CDO, 311-13
Cash-flow mapping, 239-241, 460
Cash settlement, 460
CAT bond, 391, 460
Causal relationships, 332-333
CDD, 385-386, 460
CDO, see Collateralized debt
obligation
CDO squared, 460
CDS, see Credit default swap
CDX, 303, 313-14, 460
Cheapest-to-deliver bond option, 302
Cherubini, U., 161
Chicago Board of Trade (CBOT), 27
Chicago Board Options Exchange
(CBOE), 27
Chicago Mercantile Exchange, 386
Cholesky decomposition, 151, 461
Chorafas, D. N., 339
Christofferson P. F., 211
Citron, Robert, 85, 395, 397, 398, 404
Clearinghouse, 36 ,461
Clearing margin, 461
Clewlow, L., 392
489
Clients, products & business
practices, 321, 326
Coefficient of correlation, 144
Coherent risk measure, 200, 461
Coinsurance, 336
Cole, J. B., 392
Collateral, 181-82
Collateralization, 284-85, 461
Collateralized debt obligation,
311-16, 461
cash, 311-13
equity tranche, 313
mezzanine tranche, 313
single-tranche trading, 313-15
synthetic, 313
valuation, 314
Component VaR, 207, 461
Compound option, 46, 461
Compounding frequency, 81, 461
Compounding swap, 461
Comprehensive approach (adjustment
for collateral), 181
Computer models, 357
Conditional stress testing, 213
Conditional VaR, 198, 461
Confidence level, 205-6
Confirmation, 461
Consumption asset, 461
Continuous compounding, 81-82,
461
Control areas, 388
Convenience yield, 461
Convergence arbitrage, 285, 402
Conversion factor, 170, 462
Convertible bond, 462
Convexity, 93-96, 462
Convexity adjustment, 346, 462
Cooke ratio, 169, 462
Cooling degree days, 385-86
Copula correlation, 156, 183
Copulas, 152-61, 462
and Basel II, 182-87
and calculation of VaR, 250
490
and risk aggregation, 375
and valuation of credit derivatives,
314-16
application to loan portfolios,
159-61
factor, 158-59
Gaussian, 153-56
multivariate, 158-59
Student t, 156-58
Cornish-Fisher expansion, 247-48,
462
CorporateMetrics, 196
Correlation, 143-61
definition, 144
monitoring, 146-49
vs. dependence, 144-46
Correlation smiles, 315
Cost of carry, 462
Counterparty, 462
Coupon, 84, 462
Covariance, 144, 462
Covariance rate, 146
Covered call, 462
Crashophobia, 349, 462
CRD, see Capital requirements
directive
Credit contagion, 269
Credit default swap, 299-309, 314-16,
462
add-up basket, 311
basket, 311, 314-16
binary, 307
first-to-default, 311
forward, 308-9
nth-to-default, 311
option, 308-9
spread, 300
valuation, 303-7
Credit derivatives, 187-88, 299-317,
462
Credit equivalent amount, 170-72,
462
Credit event, 299
Index
Credit exposure, 278-82
Credit indices, 303, 314
Credit rating, 255-56, 462
Credit rating transition matrix, 290,
421-22, 463
Credit risk, 255-317, 463
Credit risk economic capital, 369-70
Credit risk losses, 277-95
Credit risk migration, 290, 463
Credit risk mitigation, 283-86
Credit Risk Plus, 288-89, 369, 463
Credit substitution approach, 187
Credit trigger, 185, 285-86
Credit value at risk, 287-93, 369-70,
463
CreditGrades, 272
CreditMetrics, 196, 289-90, 369, 463
Cross gamma, 247
Crouhy, M., 192
Crude Oil, 387
Cumby, R., 138
Cumulative distribution function, 463
Currency Swap, 281-82, 410-11, 463
Cycle specific model, 369
Daiwa, 321
Damage to physical assets, 321, 327
Das, S., 317
Day count, 463
Day trade, 463
De Fountnouvelle, P., 335, 339
Deductible, 336
Default correlation, 287-94, 314-16,
463
Default intensity, 259, 265, 463
Default probability estimation,
255-73
from asset swaps, 264-65
from bond prices, 261-63
from historical data, 258-59
from credit default swaps, 306-7
from equity prices, 269-72
DeJesus-Rueff, V., 335, 339
Delbaen, F., 199, 214
Index
Delivery price, 33, 463
Delta, 56-63, 98-104, 414, 419, 463
Delta hedging, 55-63, 463
Delta neutral, 57, 59-60
Delta-neutral portfolio, 59, 70, 463
Delta neutrality, 60
Demarta, S., 162
Dependence, 144-46
DerivaGem, 59, 415, 418, 463
'
Derivative, 463
Derman, E., 76, 344, 353, 363
Deterministic variable, 463
Deutsche Bank, 373, 378, 401
Dev, A., 381
Diebold F., 356
Discount bond, 463
Discount instrument, 464
Discount rate, 464
Distance to default, 271, 464
Dividend, 464
Dividend yield, 464
Dowd, K., 214
Down-and-in option, 464
Down-and-out option, 464
Downgrade trigger, 285-86, 464
Duffle, D., 106, 214, 273
Dunbar, N., 406
Dupire, B., 353
Duration, 89-93, 95, 464
Duration matching, 464
Dynamic hedging, 71, 464
DV01, 98
EAD, see Exposure at default
Early exercise, 464
Eber, J.-M., 199, 214
Economic capital, 365-81, 464
EDF, see Expected default frequency
Efficient frontier, 6-7, 464
Efficient market hypothesis, 464
Electricity, 388-90
Electronic trading, 28, 464
Embedded option, 464
Embrechts, P., 230
491
Empirical research, 464
Employment practices and workplace
safety, 321, 326
Energy derivatives, 387-90
crude oil, 387
natural gas, 388
electricity, 388-90
Engle, R.F., 122, 125, 138, 149, 162
Enron, 286, 337, 396, 401
Equity capital, 16
Equity swap, 465
Equity tranche, 313
Equivalent annual interest rate, 81
Ergener, D., 76
Euler's theorem, 207, 377
Eurocurrency, 465
Eurodollar, 465
Eurodollar futures contract, 465
Eurodollar interest rate, 465
European Growth Trust, 373
European option, 40, 465
EWMA model, see Exponentially
weighted moving average model
Exception, 208
Excess cost layers, 390
Excess of loss reinsurance, 390
Exchange clearing house, 36
Exchange-traded market, 27, 465
Ex-dividend date, 465
Execution, delivery, and process
management, 322, 327
Exercise price, 40, 465
Exotic option, 46-48, 465
Exotics, 27
Expectations theory, 465
Expected default frequency, 272
Expected return, 2
Expected shortfall, 198-99, 200-201,
465
Expected value of a variable, 2, 465
Expiration date, 40, 465
Exponential spectral risk measure,
201
492
Index
Exponential weighting, 123, 465
Exponentially weighted moving
average model, 123-25, 147,
222-23, 465
Exposure at default, 183-84, 466
External fraud, 321, 326
External risks, 323
Extreme value theory, 224-27, 466
Eydeland, A., 392
Frachot, A., 339
Fractile, 201
French, K. R., 113, 138
Front office, 55
Froot, K. A., 393
Frye, J., 101, 252
Futures contract, 34-36, 407-8, 467
Futures option, 467
Futures price, 467
Fabozzi, F. J., 106
Face value of bond, 84
Factor, 466
Factor analysis, 466
Factor copula model, 158-59
Factor loading, 102, 466
Factor model, 151-52, 466
Factor score, 102, 466
Fama, E.F., 113, 138
Federal Deposit Insurance
Corporation (FDIC), 165
Figlewski, S., 138
Financial intermediary, 466
Finger, C. C, 296
First-to-default CDS, 311
Flannery, B.P., 130
Flavell R., 50
Flight to quality, 285
Floor, 466
Floor-ceiling agreement, 466
Floorlet, 466
Floor rate, 466
Foreign currency option, 414, 466
Floor trading, 28
Force of interest, 81
Forward contract, 32-34, 407-8, 466
Forward credit default swap, 308
Forward exchange rate, 466
Forward interest rate, 83-84, 466
Forward price, 407-8, 466
Forward rate, 83-84, 466
Forward rate agreement, 466
Foundation IRB approach, 185
Fourier transforms, 328
G-30 Policy Recommendations,
172-73, 467
Galai, D., 192
Gamma, 63-65, 104-5, 414, 419, 467
Gamma-neutral portfolio, 64, 467
GAP management, 99
GARCH(1,1) model, 125-37, 148,
222-23, 467
GARCH(p, q) model, 125
Gaussian copula model, 153-56, 250,
315-16, 467
Geczy, C, 50
Geman, H., 392, 393
Glass-Steagall Act, 165, 467
Gnedenko, D.V., 224
Gold lease rate, 58
Gordy M. B., 192
Greek letters, 55-73, 98-105, 414,
419, 467
delta, 56-63, 69-70, 98-104, 414,
419
gamma, 63-65, 69-70, 104-5, 414,
419
rho, 69-70, 414, 419
theta, 67, 69-70, 414, 419
vega, 65-66, 69, 104-5, 414, 419
Greeks, see Greek letters
Greenspan, Alan, 395
Gregory, J., 317
Gross income, 324
Grinblatt, M., 106
Guarantees, 187
Index
493
Haircut, 284, 467
Hamilton, D.T., 261
Hammersmith and Fulham, 332,
396, 404
Hanley, M., 393
Hazard rate, 259, 265, 467
Hasbrook, J., 138
HDD, 385-86, 467
Heath, D., 199, 214
Heating degree days, 385-86
Heavy-tailed distribution, 119
Hedge, 467
Hedge funds, 360, 467
Hedge ratio, 43, 467
Hedger, 467
Hedging, 43-46, 73, 100
Hendricks, D., 230
Historical default probabilities,
258-59, 265-69
Historical simulation, 217-30,
250-51, 467
Historic volatility, 115-17, 467
Holiday calendar, 468
Homogeneity, 199
Hopper, G., 214
Household International, 321
Hua P., 214
Hull, J.C., 89, 117, 222, 230, 250,
252, 263, 265, 272, 273, 308,
317, 353, 415, 419
Hunter, R., 392
Hybrid Approach 375, 468
Interest rate derivative, 468
Interest rate floor, 468
Interest rate option, 468
Interest rate risk, 79-106
Interest rate swap, 36-40, 281-82,
409-10, 468
Interest rate term structure, 79, 86
Internal controls, 406
Internal fraud, 321, 326
Internal model based approach, 176
Internal rating based approach,
182-87
advanced, 185-87, 257
foundation, 185
Internal risks, 323
International Petroleum Exchange,
387
International Swaps and Derivatives
Association, 39
In-the-money option, 40, 468
Intrinsic value, 468
Inverse floaters, 85
Investment asset, 468
Investment grade, 256
IPE, see International Petroleum
Exchange
IRB approach, see Internal ratings
based approach
ISDA, see International swaps and
derivatives association
iTraxx, 303, 313-14,468
Iben, T., 273
Implied volatility, 114-15, 468
Inception profit, 401, 468
Incremental value at risk, 206-7, 468
Initial margin, 32, 468
Instantaneous forward rate, 468
Insurance, 335-37, 359
Insurance derivatives, 390-91
Interest rate cap, 468
Interest rate collar, 468
Interest rate delta, 98-104
J.P. Morgan, 124, 195, 196
Jackson P., 214
Jamshidian, F., 249, 252
Jarrow, R.A., 421
Jett, Joseph, 345, 354, 399
Jones, F. J., 50
Jordan, J., 335, 339
Jorion, P., 106, 214, 406
Joskow, P., 392
Ju, X., 406
494
Kane, A., 139
Kani, I., 76, 353
Kealhofer, S., 273
Kendall, M. G., 220
Kendall, R., 392
Key risk indicator, 333, 469
Kidder Peabody, 345, 354, 395. 396,
399, 400
Kleinstein, A.D., 106
Kluppelberg, C, 230
KRI, see Key risk indicator
Kupiec, P., 210, 213
Kurtosis, 119, 371,469
Lando, D., 421
Laurent, J.-R, 317
Leeson, Nick, 48, 322, 373, 395, 397
Leptokurtic distribution, 119
LGD, see Loss given default
Li, D.X., 317
LIBID, see London interbank bid
rate
LIBOR, see London interbank
offered rate
LIBOR yield curve, 88, 469
LIBOR/swap yield curve, 88, 89, 469
LIBOR-in-arrears swap, 346, 469
Linear Model, 237-38, 242
Linear product, 57, 469
Liquidity-adjusted VaR, 355, 469
Liquidity black holes, 356-61, 469
Liquidity funding problems, 361
Liquidity preference theory 22, 469
Liquidity premium, 469
Liquidity risk, 354-62, 402-4, 469
Litterman, R., 273
Litzenberger, R.H., 50, 391, 393
Ljung, G. M., 133
Ljung-Box statistic, 133
Loan losses, 16
Lognormal probability distribution,
328, 469
London interbank bid rate, 87, 469
Index
London interbank offered rate, 37,
87-89, 469
Long position, 32, 42, 469
Longin, F. M., 214
Long-run variance rate, 122
Longstaff, F.A., 106
Long-Term Capital Management,
285, 360-61, 373, 396, 402
Lookback option, 47, 470
Lopez, J. A., 184, 192
Loss frequency distribution, 327
Loss given default, 183-86, 470
Loss severity distribution, 327
LTCM, see Long-Term Capital
Management
Luciano, E., 161
MA, see Maturity adjustment
Maintenance margin, 470
Margin, 31, 470
Margin call, 470
Marginal distribution, 152
Marginal value at risk, 206, 470
Mark, R., 192
Market discipline, 179, 190-91
Market maker, 28, 38, 470
Market model, 470
Market portfolio, 9, 470
Market risk, 217-52, 470
Market risk capital,
economic, 368-69
regulatory, 176-78
Marking to market, 364, 400, 470
Marking to model, 364, 400
Markowitz, H., 3, 23, 233
Marshall, C, 214
Matten, C, 380, 381
Maturity date, 40, 470
Maturity adjustment, 184
Maude, D.J., 214
Maximum-likelihood methods,
127-33, 225, 227-28, 470
McDonald, R., 392
McNeil, A. J., 162, 230
Index
Mean reversion, 127, 360, 470
Medapa, P., 330
Mercury Asset Management, 372
Merrill Lynch, 321, 372
Merton, R. C, 270, 273
Merton's Model, 269-72, 470
Mesokurtic, 119
Metallgesellschaft, 362
Mezrich, J., 130, 138, 149, 162
Mezzanine tranche, 313
Microsoft, 47, 48
Middle office, 55, 400
Mikosch, T., 230
Miller, M.H., 50
Minton, B. A., 50
Model-building approach, 233-52,
470
Model risk, 343-54, 470
Modified duration, 92, 470
Monitoring volatility, 121-33
Monotonicity, 199
Monte Carlo simulation, 248-49,
290, 328, 471
Moody's, 255, 369
Moody's KMV, 271
Moral hazard, 167, 336, 471
Multivariate GARCH models, 148
Multivariate Gaussian copula,
158-59
Multivariate normal distribution,
149-52, 471
Naked position, 471
National Association of Insurance
Commissioners (NAIC), 166
National Westminister Bank, 396,
400
Natural gas, 388
Neftci, S.N., 230
Nelken, I., 272
Nelson, D., 125, 138
Net interest income, 16, 20, 471
Net replacement ratio (NRR), 175,
471
495
Netter, J., 339
Netting, 174-75, 283, 471
New York Mercantile Exchange, 43,
387
New York Stock Exchange, 27
Ng, V., 125, 138
Noh, J., 139
Noninterest expense, 16
Noninterest income, 16
Nonlinear product, 58-63, 471
Nonsystematic risk, 9, 471
Normal distribution, 117, 150, 471
Normal market, 471
Notional principal, 37, 300, 471
Numerical procedure, 471
NYMEX, see New York Mercantile
Exchange
NYSE, see New York Stock
Exchange
Off-balance-sheet items, 170
Offer price, 28, 471
Off-peak period, 389
On-balance-sheet items, 170
On-peak period, 389
Open interest, 471
Open outcry, 471
Operational risk, 17, 188-89, 321-39,
471
Operational risk economic capital,
370
Options, 40-43, 46-48, 471
Asian
Asian basket, 48 •
at-the-money, 40
American, 40
average price call, 46
barrier, 46
basket, 46
binary, 46
call, 40
compound, 46
exotic, 46
European, 40
496
expiration date, 40
,
in-the-money, 40
lookback, 47
maturity, 40
out-of-the-money, 40
put, 40
strike price, 40
Orange County, 1, 85, 396, 397, 404,
405
OTC market, see Over-the-counter
market
Ou, S., 261
Out-of-the-money option, 40, 471
Outside-model hedging, 351
Overfitting, 353
Overparameterization, 353
Over-the-counter market, 27-28, 471
P&L decomposition, 351
Pan, J., 214
Panjer, H. H., 328
Panjer's algorithm, 328
Par value of bond, 84, 472
Par yield, 472
Parallel shift, 96, 472
Pareto distribution, 224
Partial duration, 96, 472
Partial simulation approach, 249
Pascutti, M., 50
Payoff, 472
PD, see Probability of default
Pearson, N., 406
Percentile, 201
Perraudin, W., 214
Persaud, A. D., 357, 363
Physical probability, 268
Pillar 1 of Basel II, 179
Pillar 2 of Basel II, 179, 189
Pillar 3 of Basel II, 179, 190
Plain vanilla products, 27, 30-43,
472
Platykurtic distribution, 119
Poisson distribution, 288, 328, 472
Poisson process, 472
Index
Policy limit, 336
Portfolio immunization, 96, 472
Portfolio insurance, 358, 472
Positive-semidefinite matrix, 148-49,
472
Poulsen, A., 339
Power law, 119-21, 226, 335, 472
Predescu, M., 89, 263, 265, 273
Premium, 472
Principal of bond, 84, 472
Principal components analysis,
100-104, 241, 472
Probability distribution
bivariate normal, 150, 156
bivariate Student t, 157
lognormal, 328
marginal, 152
multivariate, 149-52
normal, 150
Probability of default, 183-88,
255-73
estimation, see default probability
estimation
Procter and Gamble, 39, 48, 396,
399, 404, 405
Put-call parity, 473
Put option, 40, 473
Puttable bond, 473
Puttable swap, 473
QIS, see Quantitative impact studies
Quadratic model, 246-48, 473
Quantile, 201
Quantitative impact studies, 178
RAPM, see Risk-adjusted
performance measurement
RAROC, see Risk-adjusted return on
capital
Rating agencies, 255
Ratings transition matrix, 290
RCSA, see Risk and control self
assessment
Real-world probability, 267-69
Index
Rebalancing, 60, 473
Recovery rate, 260, 307, 473
Reference entity, 299
Regime shift, 353
Regulation, 165-92, 358
Regulatory arbitrage, 166, 473
Regulatory capital, 168-89, 203-4,
473
Reinsurance, 390
Reitano, R., 96
Remolona, E. M., 269
Repo,473
Repo rate, 473
Republic New York Corp, 321
Reset date, 473
Resti, A., 261
Retail exposures, 186-87
Reversion level, 473
Reynolds, C.E., 391, 393
Rho, 69, 414, 419, 473
Rich D., 252
Richardson, M., 221, 230
Risk-adjusted return on capital, 365,
379-81
Risk aggregation, 18
Risk and control self assessment, 333
Risk capital, 365
Risk decomposition, 18
Risk vs. return, 2-15
Risk-adjusted performance
measurement, 379
Risk-free rate, 89, 263-64, 473
Risk limits, 397-99
Risk measure, 199-202
Risk-neutral probability, 267-69
Risk-neutral valuation, 267, 473
Risk-neutral world, 474
RiskMetrics database, 124, 196
Risk-weighted amount, 169, 474
Risk-weighted assets, 169, 185, 474
Robinson, F. L., 106
Rodriguez, R. J., 273
Rogue trader risk, 323, 337
497
Roll, R., 113, 138
Roll back, 474
Roncalli, T., 339
Rosenberg, J. V., 370, 371, 375, 381
Rosengren, E., 335, 339
Ross, S., 23
Rubinstein, M., 353
Rusnak, John, 395
RWA, see Risk-weighted assets
S&P, 255, 369
Samad-Khan, A., 330
Sandor, R. L., 392
Sarbanes-Oxley Act, 337-38, 474
Scale parameter, 225
Scenario analysis, 74-75, 331, 399,
474
Schonbucher, P. J., 317
Schrand, C, 50
Schuermann, T., 356, 371, 375, 381
Scorecard approach, 334, 474
SEC, see Securities and Exchange
Commission
Second-to-default CDS, 311
Securities and Exchange
Commission, 166, 474
Shape parameter, 224
Shapiro, A., 214
Sharpe, W., 3, 23
Shih, J., 330
Short position, 30, 32, 42, 474
Short sale, 30-32, 474
Short-squeezed, 31
Simulation, 474
historical, 217-30, 250-51
Monte Carlo, 248-49, 290, 328
Sidenius, J., 317
Siegel, M., 214
Simple approach (adjustment for
collateral), 181
Single-tranche trading, 313-15
Singleton, K., 273
Sironi, A., 261
Skewness, 371-72
498
Smith, C.W., 23
Smith D. J., 40, 50
Solomon Brothers, 321
Solvency II, 167,474
Solver routine, 131, 227
Specific risk, 177, 287, 290
Specific risk charge (SRC), 177, 474
Spectral risk measure, 201, 474
Speculator, 474
Speculation, 48
Spot contract, 30
Spot interest rate, 83, 474
Spot price, 33, 474
Spot volatility, 474
Spread, credit default swap, 300
Spread changes in CreditMetrics, 292
SRC, see Specific risk charge
Standard deviation of return, 3
Standard error, 115, 221
Standardized approach
credit risk, 180-81
operational risk, 188, 324
Static hedge, 474
Static options replication, 73-74, 475
Statistically independent, 144
Stigum, M., 106
Stochastic variable, 475
Stock index, 475
Stock index futures, 475
Stock index option, 475
Stock option, 475
Storage costs, 475
Strategic risk, 370
Stress testing, 212, 399, 475
Strickland, C, 392
Strike price, 40, 475
Strip, 86, 345
Stroughair, J., 356
Structured products, 27, 58, 351-52,
475
Stuart, A., 220
Student t-copula, 156-58, 475
Student t-distribution, 156-58, 475
Index
Stulz, R. M., 23
Subadditivity, 199
Subordinated long-term debt, 15
Suo, W., 353
Supervisory review, 179, 189
Supplementary capital, 172
Swap, 36-40, 475
Swap rate, 38, 39, 88, 475
Swaption, 475
Swap yield curve, 88, 475
Swing option, 389
Synthetic CDO, 313, 475
Systematic risk, 9, 475
Systemic risk, 168, 475
Tail correlation, 157, 475
Tail loss, 198, 475
Tail value, 157
Take-and-pay option, 389
Taleb, N. N., 76
Tavakoli, J.M., 317
Taylor series expansions, 70-71, 476
Term structure of interest rates, 476
Terminal value, 476
Teukolsky, S.A., 130
Teweles, R.J., 50
Theta, 67, 69-70, 414, 419, 476
Thomson, R., 406
Tier 1 capital, 18, 172, 177, 476
Tier 2 capital, 18, 172, 177, 476
Tier 3 capital, 177, 476
Time decay, 67, 476
Time horizon, 203
Time value, 476
Total return swap, 310-11, 476
Trading book, 176
Trading day, 113, 476
Tranche, 312-13, 476
Transaction cost, 63, 476
Translation invariance, 199
Treasury bill, 85, 476
Treasury bond, 85, 476
Treasury note, 476
Treasury note futures, 476
Index
Treasury rate, 85
Tree, 417-19, 476
Turnbull, S.M., 421
Two-sided default risk, 282
Unconditional default probability,
259
Unconditional distribution, 152
Underlying variable, 476
Unilever, 372
Unsystematic risk, 12, 477
Valuation,
American options, 417-19
basket credit default swaps, 314-16
bonds, 84
collateralized debt obligations,
314-16
credit default swaps, 303-7
European options, 413-15
forward contracts, 407-8
futures contracts, 407-8
swaps, 409-11
Value at risk, 176, 195-214, 477
backtesting, 208-12
component, 207
confidence level, 205-6
credit, 287-93, 369-70
definition, 196-98
historical simulation approach,
217-30, 250-51
incremental, 206-7
marginal, 206
model building approach, 233-51
time horizon, 203
vs. expected shortfall, 198-202
Valuing American options, 417-419
Valuing European options, 413-415
Valuing forward contracts, 407-8
Valuing futures contracts, 407-8
Valuing swaps, 409-11
Van Den Brink, G.J., 340
VaR, see Value at risk
499
Variance-covariance approach, 233,
477
Variance-covariance matrix, 148
Variance rate, 112, 477
Variance targeting, 130
Varma, P., 261
Vasicek, O., 160, 162, 192
Vasicek's model, 287, 477
Vecchiato, W., 161
Vega, 65, 66, 104, 414, 419, 477
Vega neutral, 66, 477
Vetterling, W.T., 130
Vetting, 352
Volatility, 477
definition, 112
historic, 115-17
implied, 114-15
monitoring, 121-33
per day, 234
per year, 234
skew, 348
smile, 347
surface, 349
term structure, 135
Volatility models, 121-38
ARCH(m), 122
Exponentially weighted moving
average, 123, 147, 222-23
GARCH(1,1), 125-37, 222-23
GARCH(p,q), 125
Volatility per day, 234
Volatility per year, 234
Volatility skew, 348, 477
Volatility smile, 347, 477
Volatility surface, 349, 477
Volatility term structure, 135, 477
Warwick, B., 50
WCDR, see Worst-case default rate
Weather derivatives, 385-87, 477
Wei, J., 392
Wells Fargo Bank, 322
White, A., 89, 117, 222, 230, 250, 252,
263,265,272,273,308,317
500
Whitelaw, R., 221, 230
Wilmott, P., 214
Within-model hedging, 351
Worst-case default rate, 183-87
Writing an option, 42, 477
Yield on bond, 84, 477
Yield curve, 79, 477
Yield curve play, 85
Index
Young, Peter, 373
Zero, 83,
Zero-coupon bond, 477
Zero curve, 83, 478
Zero rate, 83, 478
Zhang, P. G., 406
Zhu, Y., 249, 252
Z-score, 257-58, 478